Final answer:
The probability that both groups will have the same number of men is 5/231. The statement P(A U B) < P(A) + P(B) - 1 is false.
Step-by-step explanation:
Probability of Same Number of Men in Both Groups
To find the probability that both groups will have the same number of men, we need to determine the total number of possible outcomes and the number of favorable outcomes.
There are 6 men in the group, so when dividing them into two groups of size 6, there can be (6 choose 3) ways of selecting 3 men for the first group. Once the men for the first group are chosen, the remaining 3 men will go to the second group.
Therefore, the number of favorable outcomes is (6 choose 3) = 20.
The total number of possible outcomes is the number of ways to divide 12 people into two groups of size 6, which is (12 choose 6) = 924.
Therefore, the probability that both groups will have the same number of men is 20/924 = 5/231.
True or False: P(A U B) < P(A) + P(B) - 1
False
In probability theory, the union of two events A and B is the event that either A or B or both occur. The formula P(A U B) = P(A) + P(B) - P(A ∩ B) holds true for any two events A and B.
This means that the probability of the union of two events is equal to the sum of their individual probabilities minus the probability of their intersection.
However, the formula P(A U B) < P(A) + P(B) - 1 does not hold in general. It may hold for some specific scenarios, but it is not a universally true inequality.