In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first examination is 0.8 and the probability of a the second examination is 0.7. The probability of passing at least one of them is 0.95. What is the probability of passing both?

Question

Philoid · Accepted Answer

Let E denotes the event that student passed in first examination.

And H be the event that student passed in second exam.

Given, P(E) = 0.8 and P(H) = 0.7

Also probability of passing atleast one exam i.e P(E or H) = 0.95

Or, P(E ∪ H) = 0.95

We have to find the probability of the event in which students pass both the examinations i.e. P(E ∩ H)

Note: By definition of P(A or B) under axiomatic approach(also called addition theorem) we know that:

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

∴ P(E ∪ H) = P(E) + P(H) – P(E ∩ H)

⇒ P(E ∩ H) = P(E) + P(H) – P(E ∪ H)

⇒ P(E ∩ H) = 0.7 + 0.8 – 0.95 = 1.5 – 0.95 = 0.55

∴ Probability of passing both the exams = P(E ∩ H) = 0.55