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.