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Question
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 passing 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?
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Solution
Let A and B be the events of passing first and second examinations, respectively.
P(A) = 0.8, P(B) = 0.7
Probability of passing at least one examination
= 1 – P(A’ ∩ B’) = 0.95
⇒ P(A’ ∩ B’) = 1 – 0.95
= 0.05
But A’ ∩ B’ = (A ∪ B)’ ... (By Demorgan’s Law)
∴ P(A’ ∩ B’) = P(A ∪ B)’ = 1 – P(A ∪ B)
= 0.05
∴ P(A ∪ B) = 1 – 0.05
= 0.95
Now P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
0.95 = 0.8 + 0.7 – P(A ∩ B)
P(A ∩ B) = 1.5 – 0.95
= 0.55
Thus, the probability of passing both the examinations = 0.55
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