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Question: A man and his wife appear for an interview for two vacancies in the same posts. The probability of t...

A man and his wife appear for an interview for two vacancies in the same posts. The probability of the husband’s selection is 17\dfrac{1}{7} and that of the wife’s selection is 15\dfrac{1}{5} . What is the probability that only one of them will be selected?
1)17 2)27 3)37  1)\dfrac{1}{7} \\\ 2)\dfrac{2}{7} \\\ 3)\dfrac{3}{7} \\\
4)4) None of these

Explanation

Solution

Hint : Here we will use the conditional probability and the formula to find the complementary of one event since if one is selected the other is rejected and conditional probability that if the wife is selected the husband is rejected and vice-versa.

Complete step-by-step answer :
Let A be the event that the husband is selected and
B be the event that the wife is selected.
Given that:
P(A)=17 P(B)=15  P(A) = \dfrac{1}{7} \\\ P(B) = \dfrac{1}{5} \\\
Now, the complementary probability of both the events can be given by –
P(A′)=1−P(A)P(A') = 1 - P(A)
Place the given values –
P(A′)=1−17P(A') = 1 - \dfrac{1}{7}
Take the LCM (least common multiple) in the above expression –
P(A′)=7−17P(A') = \dfrac{{7 - 1}}{7}
Simplify finding the difference of the terms in the numerator
P(A′)=67P(A') = \dfrac{6}{7} …. (A)
Similarly, P(B′)=1−P(B)P(B') = 1 - P(B)
Place the given values –
P(B′)=1−15P(B') = 1 - \dfrac{1}{5}
Take the LCM (least common multiple) in the above expression –
P(B′)=5−15P(B') = \dfrac{{5 - 1}}{5}
Simplify finding the difference of the terms in the numerator
P(B′)=45P(B') = \dfrac{4}{5} …. (B)
Now, the required probability can be given by –
P(A).P(B′)+P(A′)P(B)P(A).P(B') + P(A')P(B)
Place the known values in the above equation –
=(17).(45)+(67).(15)= \left( {\dfrac{1}{7}} \right).\left( {\dfrac{4}{5}} \right) + \left( {\dfrac{6}{7}} \right).\left( {\dfrac{1}{5}} \right)
Simplify the above expression finding the product of the terms in the numerator and the denominator.
=(435)+(635)= \left( {\dfrac{4}{{35}}} \right) + \left( {\dfrac{6}{{35}}} \right)
When denominators are equal then the numerators are combined.
=(4+635)= \left( {\dfrac{{4 + 6}}{{35}}} \right)
Perform addition in the numerator –
=(1035)= \left( {\dfrac{{10}}{{35}}} \right)
Common factors from the numerator and the denominator cancel each other.
=(27)= \left( {\dfrac{2}{7}} \right)
Hence, from the given multiple choices – the option second is the correct answer.
So, the correct answer is “Option B”.

Note : Be careful while framing the first equation and placing the values. Always remember to combine the numerators, the denominators should be the same if not the same, take LCM (least common multiple) and find its equivalent term and then simplify.