## Checkpoint - Course 2, Unit 7 Patterns in Chance

In each unit, there is a final lesson and Checkpoint that helps students summarize the key ideas in the unit. The final Checkpoint will generally be discussed in class, with the teacher facilitating the summarizing, and students making notes in their Math Toolkits (teachers may just refer to this as "notes") of any points they need to remember, adding illustrative examples as needed. If your student is having difficulty with any investigation in this unit, this Checkpoint and the accompanying answers may help you recall the concepts involved, and give you the big picture of what the entire unit is about. If your student has completed the unit, then a version of this should be in his or her notes or toolkit. Students should also have Technology Tips in their toolkits, which may be useful for this unit.

Possible Responses to Unit Summary Checkpoint
In this unit, students explored the mathematics behind waiting-time distributions. In the process, they discovered the Multiplication Rule which can be used to find P(A and B) when events A and B are independent. The bolded words are vocabulary and concepts your student should be familiar with.
a. Write a general description of a waiting-time distribution. Include how to construct the probability distribution table, what the shape of the distribution looks like, and ways to find the average waiting time.
• A 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 may be successful on the first try, or the shooter may have to wait for 10 shots for success to happen. It is important that the trials are all independent; the basketball shooter has the same chance of success on each attempt, no matter how the last trial turned out.
• Suppose the probability of success is 40% on the first attempt, then we would expect that 40% of the time you will be successful on the first trial. Of those trials that do not produce a success on the first try (60% of all trials), you can expect that 40% of those (40% of 60%) will produce a success on the second try. Of those trials which do not produce success on the first two attempts (60% of 60%), you can expect that 40% of those (40% of 60% of 60%) will produce a success on the third try. And so on. Of course, if we are producing these trials by simulation, we will probably not meet with success on the first attempt exactly 40% of the time, but the same general pattern will appear. The frequencies, and therefore, the probabilities, drop as the waiting-time gets longer. The histogram illustrates this.

• The formula for the probabilities in a waiting-time distribution is P(x) = (1 - p)x - 1(p), where x is the number of trials that had to be made to get a success, or the length of the wait, 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 trials would be P(5) = (0.6)4(0.4). This reflects the Multiplication Rule, since the probability of having to wait through 5 trials would be the same as the probability of (no and no, and no, and no, and yes), which is P(no)4P(yes).
• To find the average wait time, or expected value, you can find the mean of the frequency or probability distribution, which is or 1/p. This is also the balance point of the histogram. For example, if P(yes) = 0.4, as in the basketball example and how you try 100 different times to see how long it takes to make a basket:

Wait Time, x
(Number of Trials
to Produce a Success)
Expected Frequency, F(x) Total Number of
Trials = xF(x)
1
40
1(40) or 40 trials
2
(0.6)(40) = 24
2(24) or 24 trials
3
(0.6)(0.6)(40) = 14.4
3(14.4) or 43.2
etc.

The longer the table, the better the approximation for the expected wait time, since the number of trials can be any number from 1, 2, 3, ...

(expected wait time) = (average number trials to produce success) = (total number of trials)/(100) = (sum of this column)/100

If you stop after 3 lines of the table, you would estimate the average wait time is 1.07. If you stop the table after 10 entries, you would estimate that the average wait time is 2.4 attempts. These are underestimates of the theoretical expected wait time. (See below.)

Wait Time, x
(Number of Trials
to Produce a Success)
Probability Total "Trials"
1
0.4
1(0.4)
2
(0.6)(0.4) = 0.24
2(0.24)
3
(0.6)(0.6)(0.4) = 0.144
3(0.144)
etc.

(average wait time) = ∑ xP(x)
From the formula for the expected value of a waiting-time distribution, the average number of trials is 1/p = 1/0.4 = 2.5.
b. For what kinds of problems and under what conditions should you use the Multiplication Rule to calculate probabilities?
The Multiplication Rule is used when you want to find the probability that two or more independent events all occur.

If you would like to see specific problems from Course 2, Unit 7, a link is provided to Examples of Tasks from Course 2, Unit 7. If you are interested in following up on the Statistics strand in general, then the Scope and Sequence will help you see where different concepts are introduced. On the Statistics page, you can read an explanation of the main statistics concepts as they are developed in all four courses.