Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red balls and 12 green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let $\mathrm{E}_{1}$ be the event that the ball chosen belonged to Box I and let $\mathrm{E}_{2}$ be the event that the ball chosen belonged to Box II. Let $\mathrm{F}_{1}$ be the event that the ball chosen is red and let $\mathrm{F}_{2}$ be the event that the ball chosen is green. Then which of the following statements is (are) TRUE ?

(A) The events $\mathrm{E}_{1}$ and $\mathrm{F}_{1}$ are independent; (C) The conditional probability $P\left(F_{1} \mid E_{1}\right)$ is equal to the conditional probability $P\left(F_{1} \mid E_{2}\right)$

(B) The events $\mathrm{E}_{2}$ and $\mathrm{F}_{2}$ are dependent

(D) The conditional probability $\mathrm{P}\left(\mathrm{F}_{1} \mid \mathrm{E}_{1}\right)$ is greater than the conditional probability $\mathrm{P}\left(\mathrm{F}_{2} \mid \mathrm{E}_{2}\right)$

JEE Advanced · 2026 · 17 May 2026 · Paper 1 · Probability XII

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Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red balls and 12 green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let

E_{1}

be the event that the ball chosen belonged to Box I and let

E_{2}

be the event that the ball chosen belonged to Box II. Let

F_{1}

be the event that the ball chosen is red and let

F_{2}

be the event that the ball chosen is green.

Then which of the following statements is (are) TRUE ?

(A)The events $E_{1}$ and $F_{1}$ are independent✓ Correct
(B)The events $E_{2}$ and $F_{2}$ are dependent
(C)The conditional probability $P (F_{1} ∣ E_{1})$ is equal to the conditional probability $P (F_{1} ∣ E_{2})$ ✓ Correct
(D)The conditional probability $P (F_{1} ∣ E_{1})$ is greater than the conditional probability $P (F_{2} ∣ E_{2})$

Answer: (A), (C)

Solution

Total: 15 balls in Box I, 20 in Box II, 35 total. Red: 14 total, Green: 21 total.

P (E_{1}) = 15/35 = 3/7

P (E_{2}) = 20/35 = 4/7

P (F_{1}) = 14/35 = 2/5

P (F_{2}) = 21/35 = 3/5

P (F_{1} ∣ E_{1}) = 6/15 = 2/5

P (F_{1} ∣ E_{2}) = 8/20 = 2/5

P (F_{2} ∣ E_{2}) = 12/20 = 3/5

(A) Since

P (F_{1} ∣ E_{1}) = 2/5 = P (F_{1})

, events

E_{1}

and

F_{1}

are independent. TRUE.

(B) Since

P (F_{2} ∣ E_{2}) = 3/5 = P (F_{2})

, events

E_{2}

and

F_{2}

are independent, not dependent. FALSE.

(C)

P (F_{1} ∣ E_{1}) = 2/5 = P (F_{1} ∣ E_{2})

. TRUE.

(D)

P (F_{1} ∣ E_{1}) = 2/5

and

P (F_{2} ∣ E_{2}) = 3/5

, so

P (F_{1} ∣ E_{1}) < P (F_{2} ∣ E_{2})

. FALSE.

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JEE Advanced · 2026 · 17 May 2026 · Paper 1 · Probability XII

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