A jury pool has 19 men and 14 women, from which 12 jurors will be selected. Each person is equally likely to be chosen.

(a) How many ways are there to select 12 jurors from the jury pool?

     
Incorrect  

(b) How many ways are there to select 8 men and 4 women for the jury?

    
  

(c) What is the probability that 8 men and 4 women are selected for the jury?

     Give the probability as a fraction or as a decimal rounded to four decimal places.

    
  

(d) Use the same process as parts (a), (b), and (c) to find the probability that 9 men and 3 women are selected for the jury.

     Give the probability as a fraction or as a decimal rounded to four decimal places.

    
  

Let's solve this problem step by step.

### (a) How many ways are there to select 12 jurors from the jury pool?

The total jury pool consists of \( 19 \) men and \( 14 \) women, which means there are \( 19 + 14 = 33 \) people in total. To select \( 12 \) jurors from \( 33 \), we can use the combination formula:

\[
\text{Number of ways} = \binom{33}{12} = \frac{33!}{12!(33-12)!}
\]

Now, let's calculate \( \binom{33}{12} \).

\[
\binom{33}{12} = \frac{33 \times 32 \times \cdots \times 22}{12 \times 11 \times \cdots \times 1}
\]

### (b) How many ways are there to select 8 men and 4 women for the jury?

In this case, we need to choose \( 8 \) men from \( 19 \) and \( 4 \) women from \( 14 \). The number of ways to do this is the product of the two combinations:

\[
\text{Number of ways} = \binom{19}{8} \times \binom{14}{4}
\]

Let's calculate \( \binom{19}{8} \) and \( \binom{14}{4} \) separately, and then multiply them.

### (c) What is the probability that 8 men and 4 women are selected for the jury?

The probability is given by:

\[
P(\text{8 men, 4 women}) = \frac{\text{Number of ways to select 8 men and 4 women}}{\text{Number of ways to select 12 jurors from the pool}}
\]

From parts (a) and (b), we can plug in the values for the number of ways.

\[
P = \frac{\binom{19}{8} \times \binom{14}{4}}{\binom{33}{12}}
\]

### (d) Find the probability that 9 men and 3 women are selected for the jury.

Similarly to part (c), we need to calculate the number of ways to select 9 men from 19 and 3 women from 14, and then divide by the total number of ways to select 12 jurors from the pool.

\[
P(\text{9 men, 3 women}) = \frac{\binom{19}{9} \times \binom{14}{3}}{\binom{33}{12}}
\]

Let's now calculate the values step by step. I'll compute the necessary combinations and probabilities.Here are the results for each part of the problem:

### (a) The total number of ways to select 12 jurors from 33 people:
\[
\text{Total ways} = 354,817,320
\]

### (b) The number of ways to select 8 men and 4 women for the jury:
\[
\text{Total ways} = 75,657,582
\]

### (c) The probability that 8 men and 4 women are selected for the jury:
\[
P(\text{8 men, 4 women}) \approx 0.2132
\]
(rounded to four decimal places)

### (d) The probability that 9 men and 3 women are selected for the jury:
\[
P(\text{9 men, 3 women}) \approx 0.0948
\]
(rounded to four decimal places)

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Do you want more details on any of these steps? Here are five related questions to extend the learning:

1. How would the calculations change if the pool had an additional man or woman?
2. What is the probability of selecting exactly 7 men and 5 women?
3. How does the probability change if we wanted 10 men and 2 women instead?
4. How would you interpret these probabilities in the context of jury selection fairness?
5. Can you compute the probability of selecting at least 9 men for the jury?

**Tip**: When working with combinations in large pools, always use the combination formula to manage high computational values efficiently.

A jury pool has 19 men and 14 women, from which 12 jurors will be selected. Each person is equally likely to be chosen.

(a) How many ways are there to select 12 jurors from the jury pool?

(b) How many ways are there to select 8 men and 4 women for the jury?

(c) What is the probability that 8 men and 4 women are selected for the jury?

(d) Use the same process as parts (a), (b), and (c) to find the probability that 9 men and 3 women are selected for the jury.

Learn how to calculate the number of ways to select 12 jurors from a pool of 19 men and 14 women, along with the probability of selecting specific combinations like 8 men and 4 women. This problem involves combinatorics and probability principles. Suitable for students in Grades 10-12.

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