Trouble getting probabilites for Bayes Theorem

Question

I'm trying to think of a way to calculate the probability of P(A & B), where:

A = {a company makes me an offer} e.g. 1/20

B = {I accept the offer} e.g. 1/5

Assuming that the denominator of the Bayes Theorem will be P(A), I need to figure out P(B|A). I'm wondering:

a. if this is a number I have to make up somehow

b. or if there is a format way of calculating it.

Any tips would be useful.

Answer 1

You cannot accept an offer unless the company makes you an offer. So it is hard to understand how $P(B)$ can be larger than $P(A)$.

I think perhaps that what you are trying to say in your question is that the probability of your accepting an offer, if an offer is made, is $1$ in $5$. That is, $P(B\,|\,A)=\frac{1}{5}$.

I should think this is a question of your own priorities, and is a number you would just have to make up. For example, if you really want the job and will definitely accept any offer, then you would say $P(B\,|\,A)=1$. If you have simultaneously applied for another job that you consider equally attractive, then maybe $P(B\,|\,A)=\frac{1}{2}$. And so on. . .

Once you have decided on these numbers then you can easily calculate $$P(A\ \hbox{and}\,B)=P(B\,|\,A)P(A)\ .$$

Answer 2

Hint: $P(B|A)=\frac{P(A\cap B)} {P(A)}$

$P(A\cap B)$ is the probability that the company makes an offer and you accept it. It would be easier to find it out if you can write down the sample space, in my opinion.