2, Unit 7 - Patterns in Chance
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."
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.)
of Trials Needed
- Independent trials: The probability of a success on each trial
is unaffected by the outcome of a previous trial. For example, the
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.
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.