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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.

19.    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 \small 0.8 and the probability of passing  the second examination is  \small 0.7 . The probability of passing atleast one of them is  \small 0.95 . What is the probability of passing both? 

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Let A be the event that the student passes the first examination and B be the event that the students passes the second examination.

P(A \cup B) is probability of passing at least one of the examination.

Therefore, 

P(A \cup B) = 0.95 , P(A)=0.8, P(B)=0.7

We know,

P(A \cup B) = P(A)+ P(B) - P(A  \cap B) 

\implies P(A \cap B) = 0.8 + 0.7 - 0.95 = 1.5 -0.95 = 0.55

Hence,the probability that the student will pass both the examinations is 0.55

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