Using Counting Techniques to Determine Probabilities
The Lesson Activities will help you meet these educational goals:
· Mathematical Practices—You will make sense of problems and solve them and use mathematics to model real-world situations.
· STEM—You will apply mathematical and technology tools and knowledge to analyze real-world situations.
· 21st Century Skills—You will use critical-thinking and problem-solving skills.
Directions
You will evaluate some of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.
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Self-Checked Activities
Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.
1. Using Combinations to Find a Probability
In your problem scenario, the pool of candidate jurors consists of 26 men and 29 women. This is roughly representative of the equal numbers of males and females in the national population. From this pool, the lawyers have chosen a jury of 10 men and 2 women. Your job is to determine whether this selection could reasonably be explained by random selection. Answer the questions below to carry out your investigation.
a. What is the number of possible combinations of 10 men chosen from 26 men in the juror pool? Write your response in the form ().
Sample answer:
Ten men can be selected from 26 in ways.
b. Using symbols, what is the number of combinations of 2 women chosen from 29 women in the juror pool?
Sample answer:
Two women can be selected from 29 in ways.
c. Using symbols, express how many combinations of 10 men and 2 women are possible.
Sample answer:
Using the fundamental counting principle, the number of combinations of 10 men and 2 women is
d. What is the numerical probability of selecting a jury with 10 men and 2 women?
Sample answer:
Let E denote the event of choosing a jury with 10 men and 2 women.
The sample space for selecting 12 members from the pool contains elements. The number of ways of selecting 10 men and 2 women is.
The probability of event E =
.
The probability of selecting a jury with 10 men and 2 women is 0.005 (0.5%).
e. Does this jury composition raise a suspicion of bias? Why or why not?
Sample answer:
Yes. The choice of 10 men and 2 women from the juror pool raises a suspicion of bias because the probability of such a composition is just 0.005 (0.5%), which is very low if you compare it with other possible compositions.
f. What is the numerical probability that a randomly selected jury consists of 8 men and 4 women?
Sample answer:
Let E denote the event of choosing a jury of 8 men and 4 women.
The sample space for selecting 12 members from the pool contains elements.
The number of ways of selecting 8 men and 4 women is.
The probability of event E is
.
The probability of selecting a jury with 8 men and 4 women is 0.085 (8.5%).
g. What is the numerical probability that all the members of a randomly chosen jury would be women?
Sample answer:
Let E denote the event of choosing a jury of 12 women.
The sample space for selecting 12 members from the pool contains elements.
The numbers of ways of selecting 12 women is.
The probability of event E is
.
The probability of selecting a jury where all the members are women is 0.00012 (0.012%).
h. What is the probability that a randomly chosen jury consists of 6 men and 6 women?
Sample answer:
Let E denote the event of choosing a jury of 6 men and 6 women.
The sample space for selecting 12 members from the pool contains elements.
The number of ways of selecting 6 men and 6 women is.
The probability of event E is
.
The probability of selecting a jury with 6 men and 6 women is 0.249 (24.9%).
i. Do any of the last three jury compositions raise concerns about bias? Why?
Sample answer:
Of the three compositions, the one with 12 women raises the most concern about bias because its probability is small compared with those of the other compositions.
2. Bringing in the Addition Rule
In this activity, you will find probabilities using the addition rule.
a. What is the probability of choosing a 12-person jury with 7 or 8 women? In other words, calculate P(7 or 8 women).
Sample answer:
According to the addition rule, the probability of choosing a jury with 7 or 8 women is the sum of the individual probabilities of choosing juries with 7 women and 8 women. The probability of choosing a jury with 7 women (and 5 men) is
P(7 women)
.
The probability of choosing a jury with 8 women (and 4 men) is
P(8 women)
.
Adding the two probabilities gives the required probability:
P(7 or 8 women) = P(7 women) + P(8 women) = 0.234 + 0.146 = 0.38.
b. What is the probability that a jury chosen at random includes at least 10 men? Calculate P(at least 10 men).
Sample answer:
To find the probability of choosing a jury with at least 10 men, add the individual probabilities of choosing juries with10 men, 11 men, and 12 men and then add the individual probabilities (because each case is exclusive).
The individual probability of a jury with 10 men is P(10 men)
.
The individual probability of a jury with 11 men is P(11 men)
.
The individual probability of a jury with 12 men is P(12 men)
.
Adding the three probabilities, the probability of a jury with at least 10 men is:
P(at least 10 men) = P(10 men) + P(11 men) + P(12 men)
= 0.005 + 0.0005 + 0.00002
= 0.00552.
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