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Mathematics · 2024 · 4 marks
CBSE 2024 · Region 4 · Set 1 · Q38
Rohit, Jaspreet and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit's selection is 1/5, Jaspreet's…
Rohit, Jaspreet and Alia appeared for an interview for three vacancies in the same post. The probability of Rohit's selection is $\displaystyle \frac{1}{5}$, Jaspreet's selection is $\displaystyle \frac{1}{3}$ and Alia's selection is $\displaystyle \frac{1}{4}$. The event of selection is independent of each other.
Based on the above information, answer the following questions :
(i)
What is the probability that at least one of them is selected ?
(ii)
Find $\displaystyle \mathrm{P}(\mathrm{G} \mid \overline{\mathrm{H}})$ where G is the event of Jaspreet's selection and $\displaystyle \overline{\mathrm{H}}$ denotes the event that Rohit is not selected.
(iii)
Find the probability that exactly one of them is selected.
OR
(iii)
Find the probability that exactly two of them are selected.
Marking-scheme solution
Given $\displaystyle \mathrm{P}($ Rohit $\displaystyle )=\frac{1}{5}, \mathrm{P}($ Jaspreet $\displaystyle )=\frac{1}{3}, \mathrm{P}($ Alia $\displaystyle )=\frac{1}{4}$
(i)
P (atleast one of them is selected) $\displaystyle =1-\mathrm{P}$ (no one is selected)
$$=$\displaystyle 1$-\left(\frac{4}{5} \times \frac{2}{3} \times \frac{3}{4}\right)=\frac{3}{5}
$$(ii) $\displaystyle \mathrm{P}(\mathrm{G} \mid \overline{\mathrm{H}})=\frac{\mathrm{P}(\mathrm{G} \cap \bar{\mathrm{H}})}{\mathrm{P}(\bar{\mathrm{H}})}=\frac{1}{3}$
P (exactly one of them selected)
$$\begin{aligned}
& =\mathrm{P}(\mathrm{R}) \times \mathrm{P}(\overline{\mathrm{~J}}) \times \mathrm{P}(\overline{\mathrm{~A}})+\mathrm{P}(\overline{\mathrm{R}}) \times \mathrm{P}(\mathrm{~J}) \times \mathrm{P}(\overline{\mathrm{~A}})+\mathrm{P}(\overline{\mathrm{R}}) \times \mathrm{P}(\overline{\mathrm{~J}}) \times \mathrm{P}(\mathrm{~A})
& =\frac{6+12+8}{60}=\frac{13}{30}
\end{aligned}
P (exactly two of them selected)
$$\begin{aligned}
& =\mathrm{P}(\mathrm{R}) \times \mathrm{P}(\mathrm{~J}) \times \mathrm{P}(\overline{\mathrm{~A}})+\mathrm{P}(\mathrm{R}) \times \mathrm{P}(\overline{\mathrm{~J}}) \times \mathrm{P}(\mathrm{~A})+\mathrm{P}(\overline{\mathrm{R}}) \times \mathrm{P}(\mathrm{~J}) \times \mathrm{P}(\mathrm{~A})
& =\frac{3+2+4}{60}=\frac{3}{20}
\end{aligned}
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