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• Quests and Adventures

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The \"Cube Experimental\" is a Questmod for Fallout 3. This project places a hidden entrance, of an forgotten cube complex, to the wasteland of Washington D. C.. To explore it, you only need your current Fallout 3 character.

This mod does not have any known dependencies other than the base game.

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ATTENTION: This modification is heavily scripted to make it more intense. Don't use cheats in this mod to avoide breaking the scripts and the game. Thanks!

At beginning I had the terminals bug (no option to open doors) but I moved the cube.esm and now works. I put it just down the fallout.esm in FOMM list. Fallout.esm must to be the first in fomm list (00) and cube.esm the second (01)

Cube Experimental

Cube Experimental is a single player modification for Fallout 3, set in a long deserted, underground scientific facility right below the post-nuclear Washington D.C.

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The Rubik's cube is turning 50. This is how the iconic puzzle really works

The Rubik's Cube was created 50 years ago by Hungarian inventor Ernő Rubik.

Over 500 million of them have been sold. Needless to say, the 3D puzzle has captured the imagination of countless students around the world.

It has also inspired all kinds of competition, everything from solving blindfolded to tests of speed — and even with your feet .

But no matter the cube, the process of solving one involves math – specifically, algorithms. Roman Chavez loved Rubik's Cubes so much, he founded the Jr. Oakland Cubers in high school. Now a mathematics student at Cornell University, Roman talks to Short Wave host Emily Kwong about how to solve the cube and what life lessons he's learned from the cube.

Interested in more math episodes? Email us at [email protected] .

Listen to Short Wave on Spotify and Apple Podcasts .

Listen to every episode of Short Wave sponsor-free and support our work at NPR by signing up for Short Wave+ at plus.npr.org/shortwave .

Today's episode was produced by Hannah Chinn and edited by Rebecca Ramirez. Hannah checked the facts. Tiffany Vera was the audio engineer.

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The Rubik's cube is turning 50. This is how the iconic puzzle really works

Emily Kwong

Hannah Chinn

Rebecca Ramirez

Over 500 million Rubik's Cubes have been sold. vitranc via Getty Images hide caption

The Rubik's Cube was created 50 years ago by Hungarian inventor Ernő Rubik.

Over 500 million of them have been sold. Needless to say, the 3D puzzle has captured the imagination of countless students around the world.

It has also inspired all kinds of competition, everything from solving blindfolded to tests of speed — and even with your feet .

But no matter the cube, the process of solving one involves math – specifically, algorithms. Roman Chavez loved Rubik's Cubes so much, he founded the Jr. Oakland Cubers in high school. Now a mathematics student at Cornell University, Roman talks to Short Wave host Emily Kwong about how to solve the cube and what life lessons he's learned from the cube.

Interested in more math episodes? Email us at [email protected] .

Listen to Short Wave on Spotify and Apple Podcasts .

Listen to every episode of Short Wave sponsor-free and support our work at NPR by signing up for Short Wave+ at plus.npr.org/shortwave .

Today's episode was produced by Hannah Chinn and edited by Rebecca Ramirez. Hannah checked the facts. Tiffany Vera was the audio engineer.

Experimental Probability

The chance or occurrence of a particular event is termed its probability. The value of a probability lies between 0 and 1 which means if it is an impossible event, the probability is 0 and if it is a certain event, the probability is 1. The probability that is determined on the basis of the results of an experiment is known as experimental probability. This is also known as empirical probability.

What is Experimental Probability?

Experimental probability is a probability that is determined on the basis of a series of experiments. A random experiment is done and is repeated many times to determine their likelihood and each repetition is known as a trial. The experiment is conducted to find the chance of an event to occur or not to occur. It can be tossing a coin, rolling a die, or rotating a spinner. In mathematical terms, the probability of an event is equal to the number of times an event occurred ÷ the total number of trials. For instance, you flip a coin 30 times and record whether you get a head or a tail. The experimental probability of obtaining a head is calculated as a fraction of the number of recorded heads and the total number of tosses. P(head) = Number of heads recorded ÷ 30 tosses.

Experimental Probability Formula

The experimental probability of an event is based on the number of times the event has occurred during the experiment and the total number of times the experiment was conducted. Each possible outcome is uncertain and the set of all the possible outcomes is called the sample space. The formula to calculate the experimental probability is: P(E) = Number of times an event occurs/Total number of times the experiment is conducted

Consider an experiment of rotating a spinner 50 times. The table given below shows the results of the experiment conducted. Let us find the experimental probability of spinning the color - blue.

The experimental probability of spinning the color blue = 10/50 = 1/5 = 0.2 = 20%

Experimental Probability vs Theoretical Probability

Experimental results are unpredictable and may not necessarily match the theoretical results. The results of experimental probability are close to theoretical only if the number of trials is more in number. Let us see the difference between experimental probability and theoretical probability.

Experimental Probability Examples

Here are a few examples from real-life scenarios.

a) The number of cookies made by Patrick per day in this week is given as 4, 7, 6, 9, 5, 9, 5.

Based on this data, what is the reasonable estimate of the probability that Patrick makes less than 6 cookies the next day?

P(< 6 cookies) = 3/7 = 0.428 = 42%

b) Find the reasonable estimate of the probability that while ordering a pizza, the next order will not be of a pepperoni topping.

Based on this data , the reasonable estimate of the probability that the next type of toppings that would get ordered is not a pepperoni will be 15/20 = 3/4 = 75%

Related Sections

• Card Probability
• Conditional Probability Calculator
• Binomial Probability Calculator
• Probability Rules
• Probability and Statistics

Important Notes

• The sum of the experimental probabilities of all the outcomes is 1.
• The probability of an event lies between 0 and 1, where 0 is an impossible event and 1 denotes a certain event.
• Probability can also be expressed in percentage.

Examples on Experimental Probability

Example 1: The following table shows the recording of the outcomes on throwing a 6-sided die 100 times.

Find the experimental probability of: a) Rolling a four; b) Rolling a number less than four; c) Rolling a 2 or 5

Experimental probability is calculated by the formula: Number of times an event occurs/Total number of trials

a) Rolling a 4: 17/100 = 0.17

b) Rolling a number less than 4: 56/100 = 0.56

c) Rolling a 2 or 5: 31/100 = 0.31

Example 2: The following set of data shows the number of messages that Mike received recently from 6 of his friends. 4, 3, 2, 1, 6, 8. Based on this, find the probability that Mike will receive less than 2 messages next time.

Mike has received less than 2 messages from 2 of his friends out of 6.

Therefore, P(<2) = 2/6 = 1/3

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Practice Questions on Experimental Probability

Frequently asked questions (faqs), how do you find the experimental probability.

The experimental probability of an event is based on actual experiments and the recordings of the events. It is equal to the number of times an event occurred divided by the total number of trials.

What is the Experimental Probability of rolling a 6?

The experimental probability of rolling a 6 is 1/6. A die has 6 faces numbered from 1 to 6. Rolling the die to get any number from 1 to 6 is the same and the probability (of getting a 6) = Number of favorable outcomes/ total possible outcomes = 1/6.

What is the Difference Between Theoretical and Experimental Probability?

Theoretical probability is what is expected to happen and experimental probability is what has actually happened in the experiment.

Do You Simplify Experimental Probability?

Yes, after finding the ratio of the number of times the event occurred to the total number of trials conducted, the fraction which is obtained is simplified.

Which Probability is More Accurate, Theoretical Probability or Experimental Probability?

Theoretical probability is more accurate than experimental probability. The results of experimental probability are close to theoretical only if the number of trials are more in number.

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AP®︎/College Statistics

Course: ap®︎/college statistics   >   unit 7.

• Intro to theoretical probability
• Experimental versus theoretical probability simulation

Theoretical and experimental probability: Coin flips and die rolls

• Random number list to run experiment
• Random numbers for experimental probability
• Interpret results of simulations

Part 1: Flipping a coin

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  
• (Choice A)   Results from an experiment don't always match the theoretical results, but they should be close after a large number of trials. A Results from an experiment don't always match the theoretical results, but they should be close after a large number of trials.
• (Choice B)   Dave's coin is obviously unfair. B Dave's coin is obviously unfair.
• (Choice A)   The experimental probability got closer to the theoretical probability after more flips. A The experimental probability got closer to the theoretical probability after more flips.
• (Choice B)   The experimental probability got farther away from the theoretical probability after more flips. B The experimental probability got farther away from the theoretical probability after more flips.

Part 2: Rolling a die

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9.7 Probability

Learning objectives.

In this section, you will:

• Construct probability models.
• Compute probabilities of equally likely outcomes.
• Compute probabilities of the union of two events.
• Use the complement rule to find probabilities.
• Compute probability using counting theory.

Residents of the Southeastern United States are all too familiar with charts, known as spaghetti models, such as the one in Figure 1 . They combine a collection of weather data to predict the most likely path of a hurricane. Each colored line represents one possible path. The group of squiggly lines can begin to resemble strands of spaghetti, hence the name. In this section, we will investigate methods for making these types of predictions.

Constructing Probability Models

Suppose we roll a six-sided number cube. Rolling a number cube is an example of an experiment , or an activity with an observable result. The numbers on the cube are possible results, or outcomes , of this experiment. The set of all possible outcomes of an experiment is called the sample space of the experiment. The sample space for this experiment is { 1 , 2 , 3 , 4 , 5 , 6 } . { 1 , 2 , 3 , 4 , 5 , 6 } . An event is any subset of a sample space.

The likelihood of an event is known as probability . The probability of an event p p is a number that always satisfies 0 ≤ p ≤ 1 , 0 ≤ p ≤ 1 , where 0 indicates an impossible event and 1 indicates a certain event. A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like Table 1 .

The sum of the probabilities listed in a probability model must equal 1, or 100%.

Given a probability event where each event is equally likely, construct a probability model.

• Identify every outcome.
• Determine the total number of possible outcomes.
• Compare each outcome to the total number of possible outcomes.

Constructing a Probability Model

Construct a probability model for rolling a single, fair die, with the event being the number shown on the die.

Begin by making a list of all possible outcomes for the experiment. The possible outcomes are the numbers that can be rolled: 1, 2, 3, 4, 5, and 6. There are six possible outcomes that make up the sample space.

Assign probabilities to each outcome in the sample space by determining a ratio of the outcome to the number of possible outcomes. There is one of each of the six numbers on the cube, and there is no reason to think that any particular face is more likely to show up than any other one, so the probability of rolling any number is 1 6 . 1 6 .

Do probabilities always have to be expressed as fractions?

No. Probabilities can be expressed as fractions, decimals, or percents. Probability must always be a number between 0 and 1, inclusive of 0 and 1.

Construct a probability model for tossing a fair coin.

Computing Probabilities of Equally Likely Outcomes

Let S S be a sample space for an experiment. When investigating probability, an event is any subset of S . S . When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in S . S . Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the event and 6 possible outcomes in S , S , so the probability of the event is 4 6 = 2 3 . 4 6 = 2 3 .

Computing the Probability of an Event with Equally Likely Outcomes

The probability of an event E E in an experiment with sample space S S with equally likely outcomes is given by

E E is a subset of S , S , so it is always true that 0 ≤ P ( E ) ≤ 1. 0 ≤ P ( E ) ≤ 1.

A six-sided number cube is rolled. Find the probability of rolling an odd number.

The event “rolling an odd number” contains three outcomes. There are 6 equally likely outcomes in the sample space. Divide to find the probability of the event.

A number cube is rolled. Find the probability of rolling a number greater than 2.

Computing the Probability of the Union of Two Events

We are often interested in finding the probability that one of multiple events occurs. Suppose we are playing a card game, and we will win if the next card drawn is either a heart or a king. We would be interested in finding the probability of the next card being a heart or a king. The union of two events E and  F , written  E ∪ F , E and  F , written  E ∪ F , is the event that occurs if either or both events occur.

Suppose the spinner in Figure 2 is spun. We want to find the probability of spinning orange or spinning a b . b .

There are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is 3 6 = 1 2 . 3 6 = 1 2 . There are a total of 6 sections, and 2 of them have a b . b . So the probability of spinning a b b is 2 6 = 1 3 . 2 6 = 1 3 . If we added these two probabilities, we would be counting the sector that is both orange and a b b twice. To find the probability of spinning an orange or a b , b , we need to subtract the probability that the sector is both orange and has a b . b .

The probability of spinning orange or a b b is 2 3 . 2 3 .

Probability of the Union of Two Events

The probability of the union of two events E E and F F (written E ∪ F E ∪ F ) equals the sum of the probability of E E and the probability of F F minus the probability of E E and F F occurring together ( ( which is called the intersection of E E and F F and is written as E ∩ F E ∩ F ).

A card is drawn from a standard deck. Find the probability of drawing a heart or a 7.

A standard deck contains an equal number of hearts, diamonds, clubs, and spades. So the probability of drawing a heart is 1 4 . 1 4 . There are four 7s in a standard deck, and there are a total of 52 cards. So the probability of drawing a 7 is 1 13 . 1 13 .

The only card in the deck that is both a heart and a 7 is the 7 of hearts, so the probability of drawing both a heart and a 7 is 1 52 . 1 52 . Substitute P ( H ) = 1 4 ,   P ( 7 ) = 1 13 ,   and   P ( H ∩ 7 ) = 1 52 P ( H ) = 1 4 ,   P ( 7 ) = 1 13 ,   and   P ( H ∩ 7 ) = 1 52 into the formula.

The probability of drawing a heart or a 7 is 4 13 . 4 13 .

A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.

Computing the Probability of Mutually Exclusive Events

Suppose the spinner in Figure 2 is spun again, but this time we are interested in the probability of spinning an orange or a d . d . There are no sectors that are both orange and contain a d , d , so these two events have no outcomes in common. Events are said to be mutually exclusive events when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is

Notice that with mutually exclusive events, the intersection of E E and F F is the empty set. The probability of spinning an orange is 3 6 = 1 2 3 6 = 1 2 and the probability of spinning a d d is 1 6 . 1 6 . We can find the probability of spinning an orange or a d d simply by adding the two probabilities.

The probability of spinning an orange or a d d is 2 3 . 2 3 .

Probability of the Union of Mutually Exclusive Events

The probability of the union of two mutually exclusive events E and F E and F is given by

Given a set of events, compute the probability of the union of mutually exclusive events.

• Determine the total number of outcomes for the first event.
• Find the probability of the first event.
• Determine the total number of outcomes for the second event.
• Find the probability of the second event.
• Add the probabilities.

Computing the Probability of the Union of Mutually Exclusive Events

A card is drawn from a standard deck. Find the probability of drawing a heart or a spade.

The events “drawing a heart” and “drawing a spade” are mutually exclusive because they cannot occur at the same time. The probability of drawing a heart is 1 4 , 1 4 , and the probability of drawing a spade is also 1 4 , 1 4 , so the probability of drawing a heart or a spade is

A card is drawn from a standard deck. Find the probability of drawing an ace or a king.

Using the Complement Rule to Compute Probabilities

We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will not happen. The complement of an event E , E , denoted E ′ , E ′ , is the set of outcomes in the sample space that are not in E . E . For example, suppose we are interested in the probability that a horse will lose a race. If event W W is the horse winning the race, then the complement of event W W is the horse losing the race.

To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1.

The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is 1 9 , 1 9 , the probability of the horse losing the race is simply

The Complement Rule

The probability that the complement of an event will occur is given by

Using the Complement Rule to Calculate Probabilities

Two six-sided number cubes are rolled.

• ⓐ Find the probability that the sum of the numbers rolled is less than or equal to 3.
• ⓑ Find the probability that the sum of the numbers rolled is greater than 3.

The first step is to identify the sample space, which consists of all the possible outcomes. There are two number cubes, and each number cube has six possible outcomes. Using the Multiplication Principle, we find that there are 6 × 6 , 6 × 6 , or 36  36  total possible outcomes. So, for example, 1-1 represents a 1 rolled on each number cube.

• ⓐ We need to count the number of ways to roll a sum of 3 or less. These would include the following outcomes: 1-1, 1-2, and 2-1. So there are only three ways to roll a sum of 3 or less. The probability is 3 36 = 1 12 3 36 = 1 12
• ⓑ Rather than listing all the possibilities, we can use the Complement Rule. Because we have already found the probability of the complement of this event, we can simply subtract that probability from 1 to find the probability that the sum of the numbers rolled is greater than 3. P ( E ′ ) = 1 − P ( E )          = 1 − 1 12          = 11 12 P ( E ′ ) = 1 − P ( E )          = 1 − 1 12          = 11 12

Two number cubes are rolled. Use the Complement Rule to find the probability that the sum is less than 10.

Computing Probability Using Counting Theory

Many interesting probability problems involve counting principles, permutations, and combinations. In these problems, we will use permutations and combinations to find the number of elements in events and sample spaces. These problems can be complicated, but they can be made easier by breaking them down into smaller counting problems.

Assume, for example, that a store has 8 cellular phones and that 3 of those are defective. We might want to find the probability that a couple purchasing 2 phones receives 2 phones that are not defective. To solve this problem, we need to calculate all of the ways to select 2 phones that are not defective as well as all of the ways to select 2 phones. There are 5 phones that are not defective, so there are C ( 5 , 2 ) C ( 5 , 2 ) ways to select 2 phones that are not defective. There are 8 phones, so there are C ( 8 , 2 ) C ( 8 , 2 ) ways to select 2 phones. The probability of selecting 2 phones that are not defective is:

A child randomly selects 5 toys from a bin containing 3 bunnies, 5 dogs, and 6 bears.

• ⓐ Find the probability that only bears are chosen.
• ⓑ Find the probability that 2 bears and 3 dogs are chosen.
• ⓒ Find the probability that at least 2 dogs are chosen.
• ⓐ We need to count the number of ways to choose only bears and the total number of possible ways to select 5 toys. There are 6 bears, so there are C ( 6 , 5 ) C ( 6 , 5 ) ways to choose 5 bears. There are 14 toys, so there are C ( 14 , 5 ) C ( 14 , 5 ) ways to choose any 5 toys. C ( 6 , 5 ) C ( 14 , 5 ) = 6 2 , 002 = 3 1 , 001 C ( 6 , 5 ) C ( 14 , 5 ) = 6 2 , 002 = 3 1 , 001
• ⓑ We need to count the number of ways to choose 2 bears and 3 dogs and the total number of possible ways to select 5 toys. There are 6 bears, so there are C ( 6 , 2 ) C ( 6 , 2 ) ways to choose 2 bears. There are 5 dogs, so there are C ( 5 , 3 ) C ( 5 , 3 ) ways to choose 3 dogs. Since we are choosing both bears and dogs at the same time, we will use the Multiplication Principle. There are C ( 6 , 2 ) ⋅ C ( 5 , 3 ) C ( 6 , 2 ) ⋅ C ( 5 , 3 ) ways to choose 2 bears and 3 dogs. We can use this result to find the probability. C ( 6 , 2 ) C ( 5 , 3 ) C ( 14 , 5 ) = 15 ⋅ 10 2 , 002 = 75 1 , 001 C ( 6 , 2 ) C ( 5 , 3 ) C ( 14 , 5 ) = 15 ⋅ 10 2 , 002 = 75 1 , 001

When no dogs are chosen, all 5 toys come from the 9 toys that are not dogs. There are C ( 9 , 5 ) C ( 9 , 5 ) ways to choose toys from the 9 toys that are not dogs. Since there are 14 toys, there are C ( 14 , 5 ) C ( 14 , 5 ) ways to choose the 5 toys from all of the toys.

If there is 1 dog chosen, then 4 toys must come from the 9 toys that are not dogs, and 1 must come from the 5 dogs. Since we are choosing both dogs and other toys at the same time, we will use the Multiplication Principle. There are C ( 5 , 1 ) ⋅ C ( 9 , 4 ) C ( 5 , 1 ) ⋅ C ( 9 , 4 ) ways to choose 1 dog and 1 other toy.

Because these events would not occur together and are therefore mutually exclusive, we add the probabilities to find the probability that fewer than 2 dogs are chosen.

We then subtract that probability from 1 to find the probability that at least 2 dogs are chosen.

A child randomly selects 3 gumballs from a container holding 4 purple gumballs, 8 yellow gumballs, and 2 green gumballs.

• ⓐ Find the probability that all 3 gumballs selected are purple.
• ⓑ Find the probability that no yellow gumballs are selected.
• ⓒ Find the probability that at least 1 yellow gumball is selected.

Access these online resources for additional instruction and practice with probability.

• Introduction to Probability
• Determining Probability

9.7 Section Exercises

What term is used to express the likelihood of an event occurring? Are there restrictions on its values? If so, what are they? If not, explain.

What is a sample space?

What is an experiment?

What is the difference between events and outcomes? Give an example of both using the sample space of tossing a coin 50 times.

The union of two sets is defined as a set of elements that are present in at least one of the sets. How is this similar to the definition used for the union of two events from a probability model? How is it different?

For the following exercises, use the spinner shown in Figure 3 to find the probabilities indicated.

Landing on red

Landing on a vowel

Not landing on blue

Landing on purple or a vowel

Landing on blue or a vowel

Landing on green or blue

Landing on yellow or a consonant

Not landing on yellow or a consonant

For the following exercises, two coins are tossed.

What is the sample space?

Find the probability of tossing two heads.

Find the probability of tossing exactly one tail.

Find the probability of tossing at least one tail.

For the following exercises, four coins are tossed.

Find the probability of tossing exactly two heads.

Find the probability of tossing exactly three heads.

Find the probability of tossing four heads or four tails.

Find the probability of tossing all tails.

Find the probability of tossing not all tails.

Find the probability of tossing exactly two heads or at least two tails.

Find the probability of tossing either two heads or three heads.

For the following exercises, one card is drawn from a standard deck of 52 52 cards. Find the probability of drawing the following:

Six or seven

An ace or a diamond

A heart or a non-jack

For the following exercises, two dice are rolled, and the results are summed.

Construct a table showing the sample space of outcomes and sums.

Find the probability of rolling a sum of 3. 3.

Find the probability of rolling at least one four or a sum of 8. 8.

Find the probability of rolling an odd sum less than 9. 9.

Find the probability of rolling a sum greater than or equal to 15. 15.

Find the probability of rolling a sum less than 15. 15.

Find the probability of rolling a sum less than 6 6 or greater than 9. 9.

Find the probability of rolling a sum between 6 6 and 9 , 9 , inclusive.

Find the probability of rolling a sum of 5 5 or 6. 6.

Find the probability of rolling any sum other than 5 5 or 6. 6.

For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:

A head on the coin or a club

A tail on the coin or red ace

A head on the coin or a face card

For the following exercises, use this scenario: a bag of M&Ms contains 12 12 blue, 6 6 brown, 10 10 orange, 8 8 yellow, 8 8 red, and 4 4 green M&Ms. Reaching into the bag, a person grabs 5 M&Ms.

What is the probability of getting all blue M&Ms?

What is the probability of getting 4 4 blue M&Ms?

What is the probability of getting 3 3 blue M&Ms?

What is the probability of getting no brown M&Ms?

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting 20 20 numbers from the numbers 1 1 to 80. 80. After the player makes his selections, 20 20 winning numbers are randomly selected from numbers 1 1 to 80. 80. A win occurs if the player has correctly selected 3 , 4 , 3 , 4 , or 5 5 of the 20 20 winning numbers. (Round all answers to the nearest hundredth of a percent.)

What is the percent chance that a player selects exactly 3 winning numbers?

What is the percent chance that a player selects exactly 4 winning numbers?

What is the percent chance that a player selects all 5 winning numbers?

What is the percent chance of winning?

How much less is a player’s chance of selecting 3 winning numbers than the chance of selecting either 4 or 5 winning numbers?

Real-World Applications

Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over). 2

If you meet a U.S. citizen, what is the percent chance that the person is elderly? (Round to the nearest tenth of a percent.)

If you meet five U.S. citizens, what is the percent chance that exactly one is elderly? (Round to the nearest tenth of a percent.)

If you meet five U.S. citizens, what is the percent chance that three are elderly? (Round to the nearest tenth of a percent.)

If you meet five U.S. citizens, what is the percent chance that four are elderly? (Round to the nearest thousandth of a percent.)

It is predicted that by 2030, one in five U.S. citizens will be elderly. How much greater will the chances of meeting an elderly person be at that time? What policy changes do you foresee if these statistics hold true?

• 1 The figure is for illustrative purposes only and does not model any particular storm.
• 2 United States Census Bureau. http://www.census.gov

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Theoretical Probability & Experimental Probability

Related Pages Probability Tree Diagrams Probability Without Replacement Probability Word Problems More Lessons On Probability

In these lessons, we will look into experimental probability and theoretical probability.

The following table highlights the difference between Experimental Probability and Theoretical Probability. Scroll down the page for more examples and solutions.

How To Find The Experimental Probability Of An Event?

Step 1: Conduct an experiment and record the number of times the event occurs and the number of times the activity is performed.

Step 2: Divide the two numbers to obtain the Experimental Probability.

How To Find The Theoretical Probability Of An Event?

The Theoretical Probability of an event is the number of ways the event can occur (favorable outcomes) divided by the number of total outcomes.

What Is The Theoretical Probability Formula?

The formula for theoretical probability of an event is

Experimental Probability

One way to find the probability of an event is to conduct an experiment.

Example: A bag contains 10 red marbles, 8 blue marbles and 2 yellow marbles. Find the experimental probability of getting a blue marble.

Solution: Take a marble from the bag. Record the color and return the marble. Repeat a few times (maybe 10 times). Count the number of times a blue marble was picked (Suppose it is 6).

How to find and use experimental probability?

The following video gives another example of experimental probability.

How the results of the experimental probability may approach the theoretical probability?

Example: The spinner below shows 10 equally sized slices. Heather spun 50 times and got the following results. a) From Heather’s’ results, compute the experimental probability of landing on yellow. b) Assuming that the spinner is fair, compute the theoretical probability of landing in yellow.

Theoretical Probability

We can also find the theoretical probability of an event.

Example: A bag contains 10 red marbles, 8 blue marbles and 2 yellow marbles. Find the theoretical probability of getting a blue marble.

Solution: There are 8 blue marbles. Therefore, the number of favorable outcomes = 8. There are a total of 20 marbles. Therefore, the number of total outcomes = 20

Example: Find the probability of rolling an even number when you roll a die containing the numbers 1-6. Express the probability as a fraction, decimal, ratio and percent.

Solution: The possible even numbers are 2, 4, 6. Number of favorable outcomes = 3. Total number of outcomes = 6

Comparing Theoretical And Experimental Probability

The following video gives an example of theoretical and experimental probability.

Example: According to theoretical probability, how many times can we expect to land on each color in a spinner, if we take 16 spins? Conduct the experiment to get the experimental probability.

We will then compare the Theoretical Probability and the Experimental Probability.

The following video shows another example of how to find the theoretical probability of an event.

A spinner is divided into eight equal sectors, numbered 1 through 8. a) What is the probability of spinning an odd numbers? b) What is the probability of spinning a number divisible by 4? b) What is the probability of spinning a number less than 3?

A spinner is divided into eight equal sectors, numbered 1 through 8. a) What is the probability of spinning a 2? b) What is the probability of spinning a number from 1 to 4? b) What is the probability of spinning a number divisible by 2?

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The Rubik's cube is turning 50. This is how the iconic puzzle really works

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The Rubik’s Cube was created 50 years ago by Hungarian inventor Ernő Rubik.

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A number cube is rolled 24 times and lands on 2 four times and on 6 three times. What is the experimental probability of landing on a 2?

Here, the "event" we want to pinpoint is rolling a #2# , which occurred #4# times. There were #24# total trials, so

#"experimental probability"=4/24=color(green)(1/6#

Notice that the information about rolling a #6# is extraneous.

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Probability

A number cube is rolled 20 times and lands on 1 two times and on 5 four times. Find each experimental probability. Then compare the experimental probability to the theoretical probability. landing on 5

Answered question

A number cube is rolled 20 times and lands on 1 two times and on 5 four times. Find each experimental probability. Then compare the estimated probability to the expected probability.  landing on 5

Answer & Explanation

Ayesha Gomez

Skilled 2020-10-22 Added 104 answers

Do you have a similar question?

Recalculate according to your conditions!

New Questions in High school probability

Assume that when adults with smartphones are randomly selected, 54% use them in meetings or classes (based on data from an LG Smartphone survey). If 8 adult smartphone users are randomly selected, find the probability that exactly 6 of them use their smartphones in meetings or classes?

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In binomial probability distribution, the dependents of standard deviations must includes. a) all of above. b) probability of q. c) probability of p. d) trials.

The probability that a man will be alive in 25 years is 3/5, and the probability that his wifewill be alive in 25 years is 2/3 Determine the probability that both will be alive

How many different ways can you make change for a quarter?? (Different arrangements of the same coins are not counted separately.)

One hundred people line up to board an airplane that can accommodate 100 passengers. Each has a boarding pass with assigned seat. However, the first passenger to board has misplaced his boarding pass and is assigned a seat at random. After that, each person takes the assigned seat. What is the probability that the last person to board gets his assigned seat unoccupied? A) 1 B) 0.33 C) 0.6 D) 0.5

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You spin a spinner that has 8 equal-sized sections numbered 1 to 8. Find the theoretical probability of landing on the given section(s) of the spinner. (i) section 1 (ii) odd-numbered section (iii) a section whose number is a power of 2. [4 MARKS]

If A and B are two independent events such that P ( A ) > 0.5 , P ( B ) > 0.5 , P ( A ∩ B ¯ ) = 3 25 P ( A ¯ ∩ B ) = 8 25 , then the value of P ( A ∩ B ) is A) 12 25 B) 14 25 C) 18 25 D) 24 25

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Clinical trial tests a method designed to increase the probability of conceiving a girl. In the study, 340 babies were born, and 289 of them were girls. Use the sample data to construct a 99​% confidence interval estimate of the percentage of girls born?

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Experimental Probability

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• EPUB 3 Student View
• PDF Student View
• Thin Common Cartridge
• Thin Common Cartridge Student View
• SCORM Package
• SCORM Package Student View
• 1 - Number Cube Results
• 2 - Math Mission
• 3 - Work With Spinners
• 4 - Make Four Spinners
• 5 - Guess the Colors
• 6 - Prepare a Presentation
• 7 - Make Connections
• 8 - Spinners and Probability
• 9 - Reflect on Your Work
• View all as one page

Number Cube Results

The Number Cube interactive can show the results for rolling the number cube 30 times for each number on the number cube. Discuss the following questions.

• What do you notice about the numbers?
• How do these numbers compare with class data?

INTERACTIVE: Number Cube

Attached Resources

Number Cube  

Math Mission

Explore experimental probability using spinners.

Work With Spinners

Look at the spinner provided in the interactive.

• Predict how many times the spinner would land on each color if it were spun 100 times.     blue, red, yellow, green
• Spin the spinner 100 times. How do your actual results compare to your predictions?
• Use your results to calculate the experimental probability of each color.

INTERACTIVE: Spinner

Spinner  

File size 1.7 KB

Make Four Spinners

Using the Creating Spinners interactive, work with a partner to make spinners that will give the following results. For each spinner, predict the results for 100 spins. Then spin 100 times and use your results to find the experimental probability of each color.

• The spinner is equally likely to land on red and blue, and unlikely to land on yellow.
• The spinner is equally likely to land on red, blue, and green.
• The spinner is twice as likely to land on red as it is to land on blue, and is twice as likely to land on blue as it is to land on green.
• The spinner is certain to land on red.

INTERACTIVE: Creating Spinners

Creating Spinners  

Guess the Colors

Use the Guess the Colors interactive.

• Make a spinner that your partner cannot see.
• Spin the spinner 100 times. Tell your partner the results. Hide the colors of the spinner. Your partner should try to describe what the spinner looks like based on the results and choose each color.
• Your partner can spin more times if needed to determine what the size of the colored areas on the spinner look like.

Guess the Colors  

Prepare a Presentation

Summarize what you learned from your work with spinners. Provide examples to illustrate your conclusions.

Challenge Problem

In 1,000 spins, a spinner lands on red 257 times, yellow 170 times, blue 244 times, and green 329 times.

• If the probability of each color can be shown with a unit fraction, what does the spinner probably look like?

Make Connections

Performance task, ways of thinking: make connections.

Take notes about your classmates’ observations and explanations regarding experimental probability based on their work with spinners.

Spinners and Probability

Formative assessment, summary of the math: spinners and probability.

Write a summary about spinners and probability.

Reflect on Your Work

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

I think the difference between experimental probability and theoretical probability is…

Theoretical and Experimental Probability

Theoretical and Experimental Probability 1. A number cube is rolled 24 times and lands on 2 four times and on 6 three times. a. Find the experimental probability of landing on a 2. b. Find the experimental probability of not landing on a 6. c. Compare the experimental probability you found in part a to its theoretical probability. d. Compare the experimental probability you found in part b to its theoretical probability

0 users composing answers..

1 +0 answers.

Experimental probablility is what you 'discovered'....not what you 'expected'

a. You discovered 4 out of 24 =   4/24 = 1/6

b  You discovered three 6's out of 24 rolls.... that means 21/24= 7/8 are NOT 6

c   THEORETICAL prob would be  1/6    (4 out of 24)     compare

d   Theoretical would be 5/6    (20 out of 24)    compare

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Sticky Topics

September 2014 | By Jeroen Verfallie

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Ice Cube Says Men Can Learn a Lesson From Pepé Le Pew

The rapper/actor stars in Teenage Mutant Ninja Turtles: Mutant Mayhem —and thinks guys can learn something from animation's horniest skunk.

" I definitely will not flirt like Pepe Le Pew," he says after being asked. "He got caught up in that #MeToo stuff. Let Pepe be a lesson to all you handsy guys out there—don't do it."

Cube has been taking inspiration from wherever he can find it for rap lyrics, and presumably his on-screen character as well, for decades. So it's unsurprising that he's able to turn an off-hand question about history's horniest cartoon skunk into a bit of advice for the dudes out there.

And now Cube—real name O'Shea Jackson—gets his first chance ever to be in a major animated motion picture. He's playing the villainous Superfly in Teenage Mutant Ninja Turtles: Mutant Mayhem , a new take on the famed pizza-chomping sewer dwellers and one of the year's best received movies yet (it currently has a 95% on Rotten Tomatoes ).

"I am so good in this," he says, not long after revealing that his whole household (including son O'Shea Jr., who co-starred with Mutant Mayhem writer/producer Seth Rogen in Long Shot ) were TMNT die-hards in the '90s . "It's time to do a Superfly spinoff. It's time for Superfly to get his shine."

Gotta admit: we're on board.

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