there are 8 opposite sex married couples at a party. two people are chosen at random to win a door prize.

A. what is the probability that the 2 people will be married to each other?
B. what is the probability that the 2 people will be same sex?
C. if 6 people are chosen whats the probability that they are 3 married couples?

a) the number of ways to choose 2 out of the 16 people is C16,2) = 120

there are 8 couples married to each other, so the prob is 8/120 = 1/15

b) the number of ways that the two chosen are different is C(8,1)xC(8,1) = 64
so the number of ways that they are the same is 120-64 = 56
so prob they are the same sex = 56/120 = 7/15

Now you try c) and explain your reasoning to us

("opposite sex married couples", wouldn't Sarah Palin be proud of that textbook author, lol)

To answer these questions, we need to first find the total number of possible outcomes and then determine the number of outcomes that satisfy each condition.

Let's start with the total number of outcomes:

1. Total number of outcomes:
To choose 2 people out of 16 (8 couples), we can apply the combination formula:
Total number of outcomes = C(16, 2) = 16! / (2! * (16-2)!) = 16! / (2! * 14!) = (16 * 15) / (2 * 1) = 240.

Now, let's move on to each specific condition:

A. Probability of choosing two people who are married to each other:
Since each couple counts as one possible outcome, the number of outcomes where two people are married to each other is simply 8.

Probability = Number of favorable outcomes / Total number of outcomes = 8 / 240 = 1/30.

B. Probability of choosing two people of the same sex:
We have 8 couples, so there are 8 possible outcomes where both people are the same sex (either all males or all females).

Probability = Number of favorable outcomes / Total number of outcomes = 8 / 240 = 1/30.

C. Probability of choosing 3 married couples out of 6 people:
To choose 3 married couples out of 6 people, we need to select 6 people from the total of 12 individuals (ignoring the remaining 4 individuals who are a part of the other 4 couples).

Number of favorable outcomes = C(12, 6) = 12! / (6! * (12-6)!) = 12! / (6! * 6!) = (12 * 11 * 10 * 9 * 8 * 7) / (6 * 5 * 4 * 3 * 2 * 1) = 924.

Probability = Number of favorable outcomes / Total number of outcomes = 924 / 240 = 77/20.

So, to summarize:
A. The probability of the two people being married to each other is 1/30.
B. The probability of the two people being of the same sex is 1/30.
C. The probability of choosing 3 married couples out of 6 people is 77/20.