## Course 2, Unit 7 - Patterns in Chance

Summary
This unit introduces students to probability distributions and to the Multiplication Rule for independent events: P(A and B) = P(A) · P(B). The distribution studied is the geometric, or waiting-time, distribution. Students learn how to construct this distribution both theoretically and by simulation; then they discover how to find the average waiting time. The specific problem that motivates the unit and is studied throughout is that of rolling two dice until doubles appears. This situation occurs, for example, in the game of Monopoly when a player is sent to "jail."

Key Ideas from Course 2, Unit 7

• Waiting-time distribution (also known as a geometric distribution): Occurs in situations in which someone is watching a sequence of independent trials and waiting for a certain event to occur. For example, the trials could be a person trying to shoot baskets and waiting for success. The shooter could be successful on the first try, or the shooter might have to wait for 10 shots for success to happen. The observer records the frequency with which the event occurred on the first trial, second trial, third trial, etc., in a frequency table. (See Course 1 Unit 1 for basic work with frequency tables.)

 Number of Trials Needed Frequency 1 2 3 etc.

• Independent trials: The probability of a success on each trial is unaffected by the outcome of a previous trial. For example, the basketball shooter has the same chance of success on each attempt, no matter how the last attempt turned out.
• Graph of a waiting-time distribution: Has a characteristic shape. For example, suppose the probability of success is 40% on the first trial, then 40% of the time you will be successful on the first trial. It will take 2 trials only if the first observation was a failure and the second was a success. The probability of fail then success is (0.60)(0.40) = 0.24. The graph will look as follows:

• Formula for the probabilities in a waiting-time distribution: P(x) = (1 - p)x - 1(p), where x is the number of attempts that had to be made to get a success and p is the probability of success on any one attempt. For the above example, where the probability of a basket is 40%, the probability of having to wait through 5 attempts would be P(5) = (0.6)4(0.4).
• Multiplication Rule: States that the probability of (A and B), where A and B are independent events, is P(A) times P(B). For example, since the probability of having to wait through 5 attempts in the basketball problem would be the same as the probability of "no and no, and no, and no, and yes," which is P(no)4P(yes) = (0.6)4(0.4).
• Average wait time, or expected value: The mean of the distribution or ∑ xP(x). Since the waiting-time distribution is infinite, this is an infinite series. Using the algebra of summing series, we find that the average wait time is 1/p.