Question

23. A jury pool consists of 31 people, 19 men and 12 women. Compute the probability that a randomly selected jury of 12 people is all male.

Answer 1

We have a pool of 31 people of which 19 are men.

We have to find the probability of selecting a jury of 12 persons and all of them are male.

In this case, we are selecting from the pool and the chosen person does not return to the pool, so the composition of the pool changes.

The first place has a chance of 19 in 31 of being occupied by a man.

The second place has a chance of 18 in 30, as one of the men has been already selected: we take one unit from the count of men and also from the total.

Generalizing, the nth place will have (20-n) in (32-n) chances of being occupied by a man.

The chances of multiple independent events is equal to the product of the chances of each individual event, so we can write:

`\begin{gathered} P=\prod P_i \\ P=(19)/(31)\cdot(18)/(30)\cdot(17)/(29)\cdot(16)/(28)\cdot(15)/(27)\cdot(14)/(26)\cdot(13)/(25)\cdot(12)/(24)\cdot(11)/(23)\cdot(10)/(22)\cdot(9)/(21)\cdot(8)/(20) \\ P=0.613\cdot0.600\cdot0.586\cdot0.571\cdot0.556\cdot0.538\cdot0.520\cdot0.500\cdot0.478\cdot0.455\cdot0.429\cdot0.400 \\ P=0.000357056 \end{gathered}`

NOTE: note how the probabilities of the last places is much smaller than the first places, as there are less men to be selected in proportion to the women.

Answer: the probability that a 12-people jury is all male is P = 0.000357056.