# Probability of geting acceptance letter

Here is the problem I am currently working on:

"A high school student is anxiously waiting to receive letter telling her whether she has been accepted to a certain college. She estimates that the conditional probabilities of receiving notification on each day of next week, given that she is accepted and that she is rejected, are as follows:

• P(Monday|accepted) = .15; P(Monday|rejected) = .05
• P(Tuesday|accepted) = .20; P(Tuesday|rejected) = .10
• P(Wednesday|accepted) = .25; P(Wednesday|rejected) = .10
• P(Thursday|accepted) = .15; P(Thursday|rejected) = .15
• P(Friday|accepted) = .10; P(Friday|rejected) = .20 She estimates that her probability of being accepted is .6. "

There are currently two subproblems I am struggling with. The first one is following:

"If there is no letter through Wednesday, what is the conditional probability that she will be accepted?"

I interpreted this as "find $$P(Acc|\bar W)$$" (meaning "accepted supposing no letter came on Wednesday"). My logic is following: $$P(Acc|\bar W) = \frac{P(Acc \cap \bar W)}{P(\bar W)} = \frac{P(Acc \cap \bar W)}{1 - P(Acc \cap W) - P(Rej \cap W)} = \frac{P(\bar W | Acc ) P(Acc)}{1 - P(Acc \cap W) - P(Rej \cap W)} = \frac{P(\bar W | Acc ) \cdot 0.6}{1 - 0.25 \cdot 0.6 - 0.1 \cdot 0.4}$$ My problem is that I don't know how to find $$P(\bar W | Acc )$$ in this equation. I had the idea of equating $$P(\bar W \cap Acc ) = P(Acc) - P(Acc \cap W)$$, but this doesn't give correct answer. Eventually, I gave up and looked for the answer to the problem, which stated that $$P(\bar W \cap Acc ) = (0.6)^2$$. Since $$P(Acc) = 0.6$$, I've decided that means that $$P(\bar W | Acc) = 0.6$$, meaning that if letter didn't came on Wednesday, we had no way of assigning probability except to revert to general $$P(Acc) = 0.6$$. With that in mind, I went for the following task:

"What is the conditional probability that she will be accepted if no letter arrives that week?"

Which I interpreted as "find $$P(Acc|\bar F)$$" (because if letter didn't came on Friday, it's safe to assume it didn't came in all previous days and didn't arrive this week), and applied the same logic: $$P(Acc|\bar F) = ... = \frac{0.6^2}{1-0.1 \cdot 0.6 - 0.2 \cdot 0.4 } = \frac {6} {17} \neq \frac {9}{25}$$ which is correct answer. Because of this, it must be true that my initial assumption about that $$P(\bar W | Acc) = 0.6$$ meant was false. What I am missing to solve this problem?

(To note: correct answers to first and second problem are $$\frac {12}{27}$$ and $$\frac {9}{25}$$, respectively)

• The initial probability of her being accepted but not on Monday, Tuesday or Wednesday is $$0.6(1- 0.15-0.2-0.25) = 0.24$$
• The initial probability of her being rejected but not on Monday, Tuesday or Wednesday is $$0.4(1-0.05-0.1-0.1) = 0.3$$
So given no letter by Wednesday, her conditional probability of acceptance is $$\frac{0.24}{0.24+0.3}=\frac{12}{27}$$
• The initial probability of her being accepted but not on Monday, Tuesday, Wednesday, Thursday or Friday is $$0.6(1- 0.15-0.2-0.25-0.15-0.1) = 0.09$$
• The initial probability of her being rejected but not on Monday, Tuesday, Wednesday, Thursday or Friday is $$0.4(1-0.05-0.1-0.1-0.15-0.2) = 0.16$$
So given no letter by Friday, her conditional probability of acceptance is $$\frac{0.09}{0.09+0.16}=\frac{9}{25}$$