Question

A student appears for test $I$ and $II$. The student is successful if he passes either in test $I$ or test $II$. The probability of the student passing in test $I$ and $II$ are $p$ and $\frac{1}{3}$ respectively. If the probability that the student is successful is $\frac{1}{2}$ then find the value of $p$.

Answer 1

Given,

$P(I\cup II)=1$

Probability that the student passes the first test $P\left(I\right)=p$

Probability that the student passes the second test $P\left(II\right)=\frac{1}{3}$

Probability that the student is successful in the tests $P(I\cap II)=\frac{1}{2}$

we have,

$P(I\cup II)=P\left(I\right)+P\left(II\right)-P(I\cap II)$

$1=p+\frac{1}{3}-\frac{1}{2}$

$\Rightarrow p=1-\frac{1}{3}+\frac{1}{2}$

$\therefore p=\frac{7}{6}$