I have the following problem.
You apply for jobs and know that if you send your application then every job appointment procedure has two stages:
- You can be either invited or not invited to a personal interview and then
- You can either be selected or not selected for a job.
Assuming that you do not have any further information about the process and the selection criteria, please compute the a priori probability to get at least one job offer after sending your application to three different places.
My thoughts
I started thinking about what events are independent and which ones are dependent.
I thought that being able to get a job from place $A$ is an independent event of being able to get a job from place $B$. For this reason, I decided to consider independently the problem what is the probability of getting a job from a place $X$.
Now, the probability of being called to go to the interview (lets call this event $I$) is $\frac{1}{2}$, in other words $P(I) = \frac{1}{2}$, because we can either be called or not.
Then, the probability of getting the job (lets call this event $J$) strictly depends on event $I$, because, for example, you cannot get the job, if you don't first go to the interview.
What we actually want to know is the probability of being called and getting the job, in other words we want to know $P(I \text{ and } J)$. Since these events are dependent, we can use the rule that $P(A \text{ and } B) = P(A) \cdot P(B | A)$, where $P(B|A)$ is the probability of event $B$ happens given the fact that event $A$ has happened.
Applying this rule to my case, I need to find $P(J | I)$, because I already know $P(I) = \frac{1}{2}$. I was thinking that this probability is $\frac{1}{4}$. Why? Basically, from $\frac{1}{2}$ of the possibilities remaining we have half of the chances to get the job, so $\frac{1}{2}$ of $\frac{1}{2}$ is $\frac{1}{4}$.
I can now calculate $P(I \text{ and } J)$, which should be $P(I) \cdot P(J | I) = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8}$.
If my reasonings are correct, this should represent the probability of getting job from one place.
Since I am applying for three different jobs (which are not dependent between each other), then I have more possibilities than $\frac{1}{8}$, so I thought we could sum the possibilities of getting a job for each individual place, thus my answer would be $\frac{1}{8} + \frac{1}{8} + \frac{1}{8} = \frac{3}{8}$.
What am I doing wrong, what am I doing correct? Or can I improve something?
Best Answer
I think I find this line of reasoning rather too speculative, I'm afraid. :-)
First, you assert that $P(I) = 1/2$. I don't see any symmetry that justifies this kind of application of the principle of indifference; it's not as though the only difference between being selected for an interview and not being selected is the opportunity to go to the interview. From the job poster's perspective, they have limited time to interview candidates, whether five people apply for the posting, or five hundred.
But suppose we put that aside for the moment. You then assert that $P(J \mid I) = 1/4$, on the assumption that once half of the possibilities have been eliminated (presumably, the $\neg I$ portion), only half of the half remain, or $1/4$. But $P(J \mid I)$ is already a conditional probability—it expresses the probability of the more specific compound event $I$ and $J$ as a fraction of the probability of the condition $I$! If you are to apply the principle of indifference again, you should treat the two possibilities equally; that is, $P(J \mid I) = P(\neg J \mid I)$. And since, by excluded middle, $P(J \mid I)+P(\neg J \mid I) = 1$ necessarily, it should be the case that
$$ P(J \mid I) = P(\neg J \mid I) = \frac{1}{2} $$
Instead, you have
$$ P(J \mid I) = \frac{1}{4} $$
which implies
$$ P(\neg J \mid I) = \frac{3}{4} $$
This has no more basis than the assertion that $P(I) = 1/2$, and violates the principle of indifference to boot. (Of course, I said that the principle of indifference shouldn't be applied here, but since you appear to want to do it...)
In other words, when you arrive at $1/4$, you are determining the probability that you get the interview and you get the job (and that's $P(I, J)$), not the probability that you get the job given that you got the interview (that's $P(J \mid I)$). If you want to analyze the problem that way, there's nothing wrong with that (modulo indifference), but then you should just leave that value alone, and not multiply, again, the probability that you get the interview. That's already been accounted for in the joint probability.
I think the most we can say about it is as follows: Suppose the probability of being selected for a given job posting is $\sigma$. Then, by independence, the probability of being selected for at least one of two job postings is $1-(1-\sigma)^2$; the probability of being selected for at least one of three job postings is $1-(1-\sigma)^3$; and the probability of being selected for at least one of $k$ job postings is $1-(1-\sigma)^k$. Even independence seems rather daring, but I can see a case for it better than I can for indifference.