Final answer:
The probabilities for a jury to have an equal number of men and women and to have at least two men are calculated using combinations. For an equal split, it's (C(20,6) * C(30,6)) / C(50,12), and for at least two men, it's 1 - ((C(30,12) + C(20,1)*C(30,11)) / C(50,12)).
Step-by-step explanation:
Probability of Jury Composition
To answer these questions, we'll use combination formulas since we are dealing with selecting a group without regard to order, from a larger pool.
a) Equal number of men and women on the jury:
The total number of possible juries is the combination of 50 people taken 12 at a time: C(50,12).
For an equal number of men and women, we need 6 men and 6 women. The number of ways to choose 6 men out of 20 is C(20,6), and the number of ways to choose 6 women out of 30 is C(30,6). So, the probability is the product of these combinations divided by the total number of possible juries: (C(20,6) * C(30,6)) / C(50,12).
b) At least two men on the jury:
The probability of having at least two men is 1 minus the probability of having 0 or 1 man. The number of ways to have 0 men and all women is C(30,12), and 1 man and 11 women is C(20,1)*C(30,11). So, the probability is 1 - ((C(30,12) + C(20,1)*C(30,11)) / C(50,12)).