Question

A man and his wife appear in an interview for two posts. The probability of the husband’s selection is $\dfrac{1}{7}$ and that of his wife’s selection is $\dfrac{1}{5}$. What is the probability that only one of them will be selected?

Answer 1

Hint: According to the question to find the probability that only one of them (husband or wife) will be selected as given above first of all we have to find the probability that the husband is not selected with the help of the formula given below:
$P(\overline E ) = 1 - P(E)......................(1)$
Where, $P(E)$ is the probability of an event to occur and $P(\overline E )$ is the probability of an event not to occur.
After finding the probability that the husband is not selected $P(\overline A )$and probability that the wife is not selected we can find the probability that only one of them will be selected with the help of the formula given below:

Formula used: Required probability $ = P(A)P(\overline B ) + P(B)P(\overline A )....................(2)$

Complete step-by-step answer:
Given,
The probability of the husband’s selection $P(A) = \dfrac{1}{7}$ and,
The probability of the wife’s selection $P(B) = \dfrac{1}{5}$
Step 1: First of all we have to find probability that the husband is not selected in the interview by using the formula (1)
The probability of the husband’s selection $P(A) = \dfrac{1}{7}$
Hence,
Probability that the husband is not selected $P(\overline A ) = 1 - P(A)$
On placing the value of$P(A)$,
\[
  P(\overline A ) = 1 - \dfrac{1}{7} \\
  P(\overline A ) = \dfrac{{7 + 1}}{7} \\
  P(\overline A ) = \dfrac{8}{7} \\
 \]
Step 2: Now, we have to find probability that the Wife is not selected in the interview by using the formula (1)
The probability of the wife’s selection$P(B) = \dfrac{1}{5}$
Hence,
Probability that the husband is not selected $P(\overline B ) = 1 - P(B)$
$
  P(\overline B ) = 1 - \dfrac{1}{5} \\
  P(\overline B ) = \dfrac{{5 - 1}}{5} \\
  P(\overline B ) = \dfrac{4}{5} \\
 $
Step 3: Now to find the probability that only one of them (husband or wife) will be selected we have to use the formula (2).
On substituting all the values in the formula (2),
Probability that only one of them will be selected =
$
   = \dfrac{1}{7} \times \dfrac{4}{5} + \dfrac{1}{5} \times \dfrac{6}{7} \\
   = \dfrac{4}{{35}} + \dfrac{6}{{35}} \\
 $
On solving the obtained equation,
$
   = \dfrac{{10}}{{35}} \\
   = \dfrac{2}{7} \\
 $
So, the probability that only one of them will be selected $ = \dfrac{2}{7}$
Final Solution: Hence, by finding $P(\overline A )$(is the probability that husband is not selected), $P(\overline B )$(is the probability that wife is not selected) using the formula (1) and with the help of formula (2) we have obtained the probability that that only one of them will be selected is$ = \dfrac{2}{7}$

Note: If the probability of an event to be occur is $P(E)$then the probability of an even not be occur is $P(\overline E ) = 1 - P(E)$
Probability of an event can be expressed as proportions that range from 0 to1, and we also can express them as percentages ranging from 0% to 100%.
Probability of 0 indicates that there is no chance that a particular event will occur.
Probability of an event to occur is always in the form of $\dfrac{p}{q}$.