AP Statistics : How to find probability distributions for discrete random variables
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Example Questions
Example Question #3 : Random Variables
During a week's worth of soccer practice, a player practices 
Two steps are crucial here.
First, we need to recognize this is a binomial distribution with 
Second, we need to realize we can use a normal approximation of the binomial since 
With that said, we can calculate a 
Example Question #4 : 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?
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 #61 : Statistical Patterns And Random Phenomena
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?
The movements that this particle can make include: RRL, RLR, LRR.
The chance of getting RRL is 
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 #62 : Statistical Patterns And Random Phenomena
If you flip a biased coin, which has a 
To calculate this probability, we need to calculate the chance of getting 4 tails and then a head.
Each tail has a prob. of 
So the probability is
Example Question #3 : Random Variables
Which of the following would be considered a binomial experiment?
Rolling six dice until three of the dice show the number two
Rolling 25 dice to find the distribution of the number of spots on the faces
Predicting the probability that in a series of ten games of Rock, Paper, Scissors played with random strategy, one individual obtains six victories
Given that 36% of the population has blond hair, predicting the probability that the majority of students at a public university have blond hair
Selecting four cards from a deck in an attempt to get all of the same face (e.g., all aces)
Predicting the probability that in a series of ten games of Rock, Paper, Scissors played with random strategy, one individual obtains six victories
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 #1 : Random Variables
Which of the following is a discrete random variable?
The number of times heads comes up on 10 coin flips
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
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.
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