III. Anticipating patterns: Probability, Simulations, Random Variables

You need to be able to describe how you will perform a simulation in addition to actually doing it.

•Create a correspondence between random numbers and outcomes.

•Explain how you will obtain the random numbers (e.g., move across the rows of the random digits table, examining pairs of digits), and how you will know when to stop.

•Make sure you understand the purpose of the simulation -- counting the number of trials until you achieve "success" or counting the number of "successes" or some other criterion.

•Are you drawing numbers with or without replacement? Be sure to mention this in your description of the simulation and to perform the simulation accordingly.

If you're not sure how to approach a probability problem on the AP Exam, see if you can design a simulation to get an approximate answer.

Independent events are not the same as mutually exclusive (disjoint) events.

Two events, A and B, are independent if the occurrence or non-occurrence of one of the events has no effect on the probability that the other event occurs.

Events A and B are mutually exclusive if they cannot happen simultaneously.

Recognize a discrete random variable setting when it arises. Be prepared to calculate its mean (expected value) and standard deviation.

Recognize a binomial situation when it arises.

The four requirements for a chance phenomenon to be a binomial situation are:

1.There are a fixed number of trials.

2.On each trial, there are two possible outcomes that can be labeled "success" and "failure."

3.The probability of a "success" on each trial is constant.

4.The trials are independent.

Realize that a binomial distribution can be approximated well by a normal distribution if the number of trials is sufficiently large. If n is the number of trials in a binomial setting, and if p represents the probability of "success" on each trial, then a good rule of thumb states that a normal distribution can be used to approximate the binomial distribution if np is at least 10 and n(1-p) is at least 10.

The primary difference between a binomial random variable and a geometric random variable is what you are counting. A binomial random variable counts the number of "successes" in n trials. A geometric random variable counts the number of trials up to and including the first "success."