Question

Jurors. From 10 men and 8 women in a pool of potential jurors 12 are chosen at random to constitute a jury. Suppose that you observe the number of men who are chosen for the jury. Let A be the event that at least half of the 12 jurors are men, and let B be the event that at least half of the 8 women are on the jury.

Part (a) Determine the sample space for this experiment

Part (b) Find (A or B) ,(A & B) and (A &(not B)), listing all the outcomes for each of those three events

Part (c) Are events A and B are mutually exclusive.? are events A and (not B)? are events (not A) and (not B)? Explain.

Short answer

Part (a) $S=\{4,5,6,7,8,9,10\}$

Part (b)

$$\begin{gathered} (AorB)=\{6,7,8,9,10\} \\ (A\&B\hspace{0.17em})=(6,7,8) \\ (A(notB\left)\right)=\{9,10\} \end{gathered}$$

Part (c)

$AandB$are not mutually exclusive.

$Aand(notB)$are not mutually exclusive and

$(notA)and(notB)$are not mutually exclusive.

Step-by-step solution

Part (a) Step 1. Given information.

A jury is formed by selecting 12 possible jurors at random from a pool of 10 men and 8 women.

Part (a) Step 2.  Sample space for the investigation at hand

All possible outcomes are represented in the sample space. A jury of 12 people is required, thus at least 4 men will be chosen out of the total of 8 women.

$S=\{4,5,6,7,8,9,10\}$

Part (b) Step 1.  The various consequences of the events ( A or B), (A & B) and (A (not B))

A is a event in which at least half of the jury members are male.

$A=\{6,7,8,9,10\}$

B is an event in which at least half of the eight women on the jury are female. As a result, the jury will consist of a maximum of 8 men and a minimum of 6.

$B=\{6,7,8\}$

Now

role="math" localid="1651472188677" $$\begin{gathered} (AorB)=\{6,7,8,9,10\} \\ (A\&B\hspace{0.17em})=(6,7,8) \\ (A(notB\left)\right)=\{9,10\} \end{gathered}$$

Part (c) Step 1. the events , A and B, A and (not B), (not A) and (not B) are mutually exclusive.

A jury is formed by selecting 12 possible jurors at random from a pool of 10 men and 8 women.

$$\begin{gathered} A\cap B\ne \varphi \\ A\cap B^{\text{'}}\ne \varphi \\ {A}^{\text{'}}\cap B^{\text{'}}\ne \varphi \end{gathered}$$

$AandB$are not mutually exclusive.

$Aand(notB)$are not mutually exclusive and

$(notA)and(notB)$are not mutually exclusive.