Question

In a committee of $25$ member, each member is proficient either in mathematics or in statistics or in both. If $19$ of these are proficient in mathematics. $16$ in statistics, find the probability that a person selected from the committee is proficient in both.

Answer 1

Solution:

Let the $A$ be the event that the person is proficient in mathematics, and $B$ be the event that the person is proficient in statistics and $S$ be the sample.

$\therefore P\left(A\right)=\frac{19}{25},P\left(B\right)=\frac{16}{25}$

Since each member is proficient either in mathematics or in statistics or in both.

$\therefore A\cup B=S⟹P(A\cup B)=P\left(S\right)=1.$

By addition theorem on probability we have,

$P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

or, $P(A\cap B)=P\left(A\right)+P\left(B\right)-P(A\cup B)$

or, $=\frac{19}{25}+\frac{16}{25}-1$

or, $=\frac{35-25}{25}$

or, $=\frac{10}{25}$

or, $=\frac{2}{5}$

$\therefore P(A\cup B)=\frac{2}{5}$