# AP Statistics : Random Variables

## Example Questions

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### Example Question #1 : How To Simulate Random Behavior

Which of the following is NOT a discrete random variable?

Time taken to watch the first 4 seasons a TV show

Number of midterms taken in a 10-week course

All of these answers are discrete variables

Amount of money in a savings account

Number of lip products in a girl's makeup bag

Time taken to watch the first 4 seasons a TV show

Explanation:

By definition, a discrete random variable is a random variable whose values can be "counted" one by one. A continuous random variable is a random variable that can take any value on a certain interval. Of these choices, the number of lip products, the amount of money, and the number of midterms taken are all discrete random variables, as the respective values can be counted; however, the time taken to watch the first four seasons of a TV show is a continuous random variable, as not everyone will take the same amount of time to watch all those episodes (i.e. some might fastf-orward/replay parts of episodes).

### Example Question #2 : How To Simulate Random Behavior

Police estimate that 85% of drivers do not text and drive. They set up a safety roadblock at a busy intersection to check for this infraction.

What is the probability that the first texter is in the thirteenth car stopped?

Explanation:

This question pertains to the concept of Geometric Probability Distribution, which states that the probability of  trials until the first success is as follows:

In this equation,  is the probability of success (in this case, a driver caught texting), and  is the probability of failure.

In this particular problem, , and . To calculate the probability of finding that the first driver caught texting is in the thirteenth car stopped, we can calculate . The final answer is .

### Example Question #1 : How To Find Probability Distributions For Discrete Random Variables

During a week's worth of soccer practice, a player practices  total free kicks and has a  chance of scoring. What is the probability that he or she scored at least  times? Assume each shot is independent.

Explanation:

Two steps are crucial here.

First, we need to recognize this is a binomial distribution with  and .

Second, we need to realize we can use a normal approximation of the binomial since  and , which are both larger than 5.

With that said, we can calculate a -score and its -value, keeping in mind that our mean will be  and our standard deviation will be , which is about .

### Example Question #1 : Random Variables

Suppose you are throwing three darts and you have a one third chance of hitting the bull's eye. Each throw is independent of one another. What is the chance of hitting the bull's eye at least once?

Explanation:

To calculate Prob(at least one bull's eye), we can instead compute one minus the complementary probability, P(no bull's eye).

So we have P(at least one bull's eye)=1-P(no bull's eye).

The chance of getting no bull's eyes is .

This means the probability of getting at least one bull's eye is

### Example Question #2 : Random Variables

A particle travels left with probability one sixth and right with probability five sixths. Each movement is independent of the others. What is the chance that after three movements, the particle ends up one unit to the right?

Explanation:

The movements that this particle can make include: RRL, RLR, LRR.

The chance of getting RRL is . This is also the chance of getting any of those movements.

To get the total probability, we can add up the individual probabilities since the events are all mutually exclusive.

Thus, we get the following as the solution.

### Example Question #4 : How To Find Probability Distributions For Discrete Random Variables

If you flip a biased coin, which has a  chance of being heads and  of being tails, until you get a head, what is the chance that it takes five flips until you get a head?

Explanation:

To calculate this probability, we need to calculate the chance of getting 4 tails and then a head.

Each tail has a prob. of  and a head is , so we multiply  to the power of 4 (because we need 4 tails) by  (for the single head).

So the probability is

.

### Example Question #1 : Random Variables

Which of the following would be considered a binomial experiment?

Predicting the probability that in a series of ten games of Rock, Paper, Scissors played with random strategy, one individual obtains six victories

Selecting four cards from a deck in an attempt to get all of the same face (e.g., all aces)

Rolling 25 dice to find the distribution of the number of spots on the faces

Rolling six dice until three of the dice show the number two

Given that 36% of the population has blond hair, predicting the probability that the majority of students at a public university have blond hair

Predicting the probability that in a series of ten games of Rock, Paper, Scissors played with random strategy, one individual obtains six victories

Explanation:

There are four conditions that need to be satisfied for a binomial experiment:

1) Each trial must have two outcomes.

2) Each trial must be independent.

3) All trials must be identical.

4) The probabilities of the outcomes remain constant must not change with each trial.

The only choice that satisfies all four of these conditions (and is therefore a binomial experiment) is the rock-paper-scissors scenario.

### Example Question #6 : How To Find Probability Distributions For Discrete Random Variables

Which of the following is a discrete random variable?

The amount of water that passes through a dam in a random hour

The rate of return on a random stock investment

The length of a random caterpillar

The number of times heads comes up on 10 coin flips

The number of times heads comes up on 10 coin flips

Explanation:

A discrete variable is a variable which can only take a countable number of values. For example, the number of times that a coin can come up heads in ten flips can only be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10. Thus, there are a countable number of possible outcomes (in this case 11). This is true for coin flips, but not for caterpillar length, water flow, or rates of return for stocks.

### Example Question #1 : Random Variables

Let us suppose you are a waiter. You work your first four shifts and receive the following in tips: (1) 20, (2) 30, (3) 15, (4) 5. What is the mean amount of tips you will receive in a given day?

Explanation:

The answer is 17.5. Simply take the values for each day, add them, and divide by the total number of days to obtain the mean:

### Example Question #2 : Random Variables

There are  collectable coins in a bag.  are  ounces,  are  ounces,  are  ounces, and  are  ounces. If one coin is randomly selected, what is the mean possible weight in ounces?