Four people are chosen at random. What is the probability that:
(a) One of the first three people chosen has their birthday in the same month as the fourth person?
The solution given for this question is just the complement of
$P(\text{None of the first three in same month as 4th})= 1- (\frac {11}{12})^3 $
I feel that the answer is wrong here because the complement of "None of the first three in same month as 4th" doesn't equal to "exactly one of the first three people chosen has their birth day in the same month" but rather the complement means that not all of the first three is in the same month as 4th?
Therefore, shouldn't the answer be $\binom {3}{1} \frac {1}{12} \cdot\ \frac {11}{12} \cdot\ \frac {11}{12}$
(b) The first and second people chosen have their birthdays in the same month, given that there is some pair of people having their birthdays in the same month?
For (b) the answer given was P(first and second in same month)/P(some pair in same month)
In particular, the P(some pair in same month) I get that it's easier to use rule of complement to get the answer: $1-(\frac {12}{12} \frac {11}{12}\frac {10}{12}\frac {9}{12} ) $
However, how come my answer here is wrong:
$\binom {4}{2} \frac {12}{12} \frac {1}{12} \frac {11}{12} \frac {10}{12}$
My rationale is, there are $4$ items in total, we are picking $2$ at a time to make them a pair thus the combination should be $4C2$.
The next step is to figure out the probability. Assume that the 1st person can be any of the month $\frac {12}{12}$. Assuming the 2nd person share the same birthday as the 1st then it should be $\frac {1}{12}$. This completes the first pair therefore, the 3rd person has a probability of $\frac {11}{12}$ and subsequently the 4th person's probability is $\frac {10}{12}$
What is wrong with my reasoning here?
Many thanks in advance!
I think the issue in both of these is the same. In the first part, your solution assumed the question meant "exactly one", and the other solution assumed it meant "at least one". (Here I think the question is genuinely unclear.)
In the second part you are assuming that exactly one pair of people are born in the same month. (I think you have the probability of this happening correct.) However, the question says "there is some pair ...", which means at least one such pair exists. Therefore they are working out the probability of this happening as the complement of the probability that all four are born in different months. (Here I think the question is clear, and you have misinterpreted it.)