A company has hired 3 programmers, who are tasked with building a piece of software. The work is divided between the programmers is 25%, 35%, and 40%. They're asked to provide project price estimations.
Ei(i=1,2,3) - is defined as the estimation given by programmer (i).
Each programmer may make an error in calculating their estimation, at the probabilities: 0.017, 0.031, and 0.045 respectively. If an estimation error has occurred, what is the probability that each engineer was responsible for this?
Adapted from a textbook. In order to address this equation, i'd simply calculate the probability that any given engineer is 'selected' i.e. 25%/35%/40% and the probability that, given they are 'selected', how likely are they to make a mistake. For example:
P(A) = 25% - engineer (i)
P(A∩B) = 1.7% - engineer (i)'s error rate
P(A|B) = 0.017/0.25 = 0.068 = 6.8%
However this ignores the information given as Ei(i=1,2,3) - I am not sure what this denotes, or what to do with it. Can anybody help?
EDIT:
So, it is my understanding that P(A|B) would describe "given that a specific programmer is chosen, what is the probability that he will make an error?". This is not what the question asks. Instead it says "given that an error has occurred...." so we need to invert the conditional probability to calculate P(B|A), as according to Bayes' theorem:
P(B|A) = P(A|B) * P(B)/P(A)
For the first programmer:
P(A|B) = 0.017/0.25 = 0.068
P(B) = 0.017
P(A) = 0.25
P(B|A) = 0.068*(0.017/0.25) = 0.004624
This probability, however, seems remarkably low.
What am I doing wrong?
It looks like it's $E_i$ for $i=1$, or $i=2$, or $i=3$. i.e. the "estimation" $E_1$ for programmer 1, $E_2$ for programmer 2, and so on. It's not a function. (In the textbook, I assume that the $i$ is a subscript to $E$, and there is probably a space between $E_i$ and the left parenthesis.) I don't see why it's important, though. That notation plays no role in the problem you have stated.
For the rest of it, notice that it says "If [i.e. given that] an estimation error has occurred." You will need to use Bayes theorem.
Heuristic: You're calculating the probability of a cause (programmer) given its effect (an error). That often needs Bayes theorem, although it depends what you're given.
(If you're not sufficiently familiar with Bayes theorem then have fun learning about it! It's important and valuable, and not very difficult.)
It's not clear to me what $A$ and $B$ stand for, but I think you have enough of a start with those tips for you to figure it out now.
EDIT in response to OP's edit:
Given the way that you have described the problem, it sounds like 0.017 is itself the conditional probability of an error given that it's programmer #1 who did the work, i.e. $P(A|B)=0.017$.
"Each programmer may make an error in calculating their estimation, at the probabilities ....." Think about it this way: "Each programmer may make an error at probability" means "for a given programmer, they may make an error at probability", i.e. given that it is programmer $i$, the error rate is .... (There are more detailed comments about this down below.)
You have made progress in understanding--good--but you'll need to study Bayes theorem a bit more. Don't assume that all of the quantities needed on the right hand side of Bayes theorem are stated explicitly in the problem. Look at the more complicated versions of Bayes theorem, too.
(About your final calculation, by the way: What you wrote is the square of 0.017/0.25. If you notice that, it should seem wrong, since if that's what Bayes theorem was supposed to do, it would be stated differently--as a square of a ratio.)
More on interpreting the statement about error rates:
"at the probabilities" doesn't explicitly say what is "at" those probabilities, and the first part of the sentence just describes programmers making errors. So I can understand you thinking that this is the probability of both events rather than the conditional probability. However, it seems clear to me that the phrasing of the sentence describes the conditional probability. If it's not conditional probability, then there is not enough information to apply Bayes theorem, as far as I can see.