a class of 60 students, 15 students failed in exam A, 25 students failed in exam B ,8 students failed in both , what is the probability of a student passing A and failing B ?

when I solve it using Venn diagrams the probability is 17/60

but when I solve it using P(Passing A & failing B)=P(A' intersection B)=P(A')*P(B) the result =18.75/60

shouldn't the two events be mutually exclusive (independent) ,passing A shouldn't affect passing B and vice versa so the intersection equals their product, so what is it that I'm doing wrong ?

  • $\begingroup$ I think you mean to ask either "what proportion of students passed exam A but failed B" or "if a student is selected uniformly at random, what is the probability that he will have passed A and failed B". We have no business predicting the future. $\endgroup$ – nullUser Jun 5 '13 at 14:29
  • $\begingroup$ @nullUser ,Ross Millikan &amWhy :thank you. $\endgroup$ – Oz Le Jun 5 '13 at 15:10

Mutually exclusive and independent are two different things. Mutually exclusive means they cannot both happen, but one can certainly pass A and fail B. Independent here means the probability of passing A is the same for those students that passed B as those that failed B. We are not given that. Your Venn diagram solution is correct, while the second formula assumes independence, which is not the case here.


More than half the students who failed test A also failed test B. But less than $1/3$ of the students who failed B, also failed A. So performance in A is not independent of performance in B. For that reason, we cannot use the "formula" you tested, would be valid to use only if the events were independent.

Your Venn Diagram result calculation is, however, valid and correct.

  • $\begingroup$ Deserves a TU! +1 $\endgroup$ – Amzoti Jun 6 '13 at 0:27
  • $\begingroup$ @Amzoti Tell me about it! I remember my first research paper: using index cards to keep references, quotes, paraphrased comments...for structuring a research paper! And...I remember the good old typewriter (and white-out!), Hah! Kids now'days have it soooooo easy ;-) $\endgroup$ – Namaste Jun 6 '13 at 0:41
  • $\begingroup$ @Amzoti Oh my, nostalgia! I can remember it so well! $\endgroup$ – Namaste Jun 6 '13 at 0:47

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