Question

In a committee of 25 members, 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

Hint: To solve the given question, first we will find the probability of each of the subjects separately with respect to the total members in a committee. And then we will find the union of both the given subjects.

Complete step by step 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.
$\therefore P(A) = \dfrac{{19}}{{25}}$ , $P(B) = \dfrac{{16}}{{25}}$
Since each member is proficient either in mathematics or in statistics or in both.
$
  \therefore A \cup B = S \\
   \Rightarrow P(A \cup B) = P(S) = 1 \;
 $
By addition theorem on probability we have,
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
or, $P(A \cap B) = P(A) + P(B) - P(A \cup B)$
$
   = \dfrac{{19}}{{25}} + \dfrac{{16}}{{25}} - 1 \\
   = \dfrac{{35 - 25}}{{25}} \\
   = \dfrac{{10}}{{25}} \\
   = \dfrac{2}{5} \;
 $
$\therefore P(A \cup B) = \dfrac{2}{5}$
Hence, the probability that a person selected from the committee is proficient in both mathematics and statistics is $\dfrac{2}{5}$ .
So, the correct answer is “$\dfrac{2}{5}$”.

Note: The actual outcome is considered to be determined by chance. The word probability has several meanings in ordinary conversation. Two of these are particularly important for the development and applications of the mathematical theory of probability.