Let $E_{1}$ and $E_{2}$ be two events such that the conditional probabilities $\mathrm{P}\left(\mathrm{E}_{1} \mid \mathrm{E}_{2}\right)=\frac{1}{2}$, $\mathrm{P}\left(\mathrm{E}_{2} \mid \mathrm{E}_{1}\right)=\frac{3}{4}$ and $\mathrm{P}\left(\mathrm{E}_{1} \cap \mathrm{E}_{2}\right)=\frac{1}{8}$. Then:

(C) $\mathrm{P}\left(\mathrm{E}_{1} \cap \mathrm{E}^{\prime}{ }_{2}\right)=\mathrm{P}\left(\mathrm{E}_{1}\right) \cdot \mathrm{P}\left(\mathrm{E}_{2}\right)$

(A) $\mathrm{P}\left(\mathrm{E}_{1} \cap \mathrm{E}_{2}\right)=\mathrm{P}\left(\mathrm{E}_{1}\right) \cdot \mathrm{P}\left(\mathrm{E}_{2}\right)$

(B) $\mathrm{P}\left(\mathrm{E}_{1}^{\prime} \cap \mathrm{E}^{\prime}{ }_{2}\right)=\mathrm{P}\left(\mathrm{E}_{1}{ }^{\prime}\right) \cdot \mathrm{P}\left(\mathrm{E}_{2}\right)$

(D) $\mathrm{P}\left(\mathrm{E}_{1}^{\prime} \cap \mathrm{E}_{2}\right)=\mathrm{P}\left(\mathrm{E}_{1}\right) \cdot \mathrm{P}\left(\mathrm{E}_{2}\right)$

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Probability

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Let

E_{1}

and

E_{2}

be two events such that the conditional probabilities

P (E_{1} ∣ E_{2}) = \frac{1}{2}

P (E_{2} ∣ E_{1}) = \frac{3}{4}

and

P (E_{1} \cap E_{2}) = \frac{1}{8}

. Then:

(A) $P (E_{1} \cap E_{2}) = P (E_{1}) \cdot P (E_{2})$
(B) $P (E_{1}^{'} \cap E^{'}_{2}) = P (E_{1}^{'}) \cdot P (E_{2})$
(C) $P (E_{1} \cap E^{'}_{2}) = P (E_{1}) \cdot P (E_{2})$ ✓ Correct
(D) $P (E_{1}^{'} \cap E_{2}) = P (E_{1}) \cdot P (E_{2})$

Answer: (C)

Solution

From

P (E_{2} ∣ E_{1}) = \frac{3}{4}

and

P (E_{1} \cap E_{2}) = \frac{1}{8}

, we get

P (E_{1}) = \frac{P ( E _{1} \cap E _{2} )}{P ( E _{2} ∣ E _{1} )} = \frac{1/8}{3/4} = \frac{1}{6}

. From

P (E_{1} ∣ E_{2}) = \frac{1}{2}

, we get

P (E_{2}) = \frac{P ( E _{1} \cap E _{2} )}{P ( E _{1} ∣ E _{2} )} = \frac{1/8}{1/2} = \frac{1}{4}

. Thus

P (E_{1}) \cdot P (E_{2}) = \frac{1}{6} \cdot \frac{1}{4} = \frac{1}{24}

. Now

P (E_{1} \cap E_{2}^{'}) = P (E_{1}) - P (E_{1} \cap E_{2}) = \frac{1}{6} - \frac{1}{8} = \frac{1}{24}

, which equals

P (E_{1}) \cdot P (E_{2})

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Single correct option

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JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Probability

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