Problem: In a school, $30\%$ of students have glasses. $20\%$ of students with glasses play sports. $60\%$ of students without glasses play sports. If we randomly choose a student, find probability that a student without glasses (chosen randomly) plays a sport.
My Approach
- $P(O)=0.3$ is the probability that a student has glasses.
- $P(S|O) = 0.2$ is the probability that a student with glasses plays sport (P of S given O).
- $P(S|O^c)=0.6$, where $O^c$ is {Sample Space – $O$}. The probability that a student without glasses plays sport $(P(S\text{ given }O^c)$.
I have to find $P(O^c|S)$ (the opposite of $P(S|O^c)$, meaning "a student plays sport without glasses").
I know that $80\%$ of students play a sport, and that $70\%$ of students don't have glasses. So in order to solve the problem I have to do $$P(O^c|S) = \frac{0.8 \times 0.7}{0.8} = 0.7 $$ but given correct solution is $0.87$ $(0.7\neq0.87)$.
EDIT: I think the problem is in this assumption: I've assumed that $0.8$ and $0.7$ were two independent probabilities, therefore I've multiplied $0.8$ with $0.7$. If this is not correct, then I don't know how to find probability of intersection.
Best Answer
I disagree with the OP's (i.e. original poster's) interpretation of the problem. This implies that I also disagree with the analysis in the answer of SacAndSac. It also implies that I disagree with the answer that was given as correct, namely $0.875$.
The reason that I disagree is because of how the problem is worded. The statement:
is interpreted by me to indicate that you are to confine your focus to only those students that are without glasses, and to then determine what fraction of them play a sport.
Certainly, there are problems with my interpretation:
This implies that the information that I claim is being asked for has already been directly handed to you.
The real difficulty here is that although sometimes the OP will misinterpret a problem, I don't think that is the case here. The impression that I got is that the OP faithfully quoted the problem composer's wording.
So what you have is a problem whose wording makes the problem a cross between trivial and nonsensical. So the MathSE reviewers and the OP are reversing the interpretation into a sensible problem.
That is all fine, except that that isn't the problem that the problem composer worded, assuming that the OP faithfully presented the problem's wording.