Question

In an entrance test that is graded on the basis of two examination, 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?

Answer 1

Let A be the event that the student passes the first examination and B be the event taht the student passes the second examination.
Then $P\left(A\right)=0.8,P\left(B\right)=0.7$
and $P(a\cup B)=0.95$
We know that
$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cup B)\therefore 0.95=0.8+0.7-P(A\cup B)\therefore 0.95=1.5-P(A\cap B)\therefore P(A\cap B)=1.5-0.95=0.55$