GATE 2026 IN Q.36: solution and answer

GATE 2026 · INQ.36 · 2 marks · MCQ

A box contains three apples and two oranges. Two fruits are removed randomly in succession. The probability that the first is an apple and the second is an orange is ______ .

(A)
$\frac{3}{5}$
(B)
$\frac{3}{20}$
(C)
$\frac{3}{10}$
(D)
$\frac{2}{5}$

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Answer and solution

ANSWER

The probability that the first fruit is an apple and the second is an orange is $\frac{3}{5} \cdot \frac{2}{4} = \frac{3}{10}$ , which is option (C).

Official answer: CSource: Official answer key, GATE 2026 IN (IIT Guwahati)

The box holds $3$ apples and $2$ oranges, $5$ fruits in all. "Removed in succession" means the first fruit is not put back before the second is drawn, so this is drawing without replacement. Let $A_{1}$ be "first fruit is an apple" and $O_{2}$ be "second fruit is an orange".
First draw: $P (A_{1}) = \frac{3}{5}$ .
Once an apple is out, $4$ fruits are left, and all $2$ oranges are still among them. So $P (O_{2} ∣ A_{1}) = \frac{2}{4} = \frac{1}{2}$ .
Multiplication rule for dependent events: $P (A_{1} \cap O_{2}) = P (A_{1}) P (O_{2} ∣ A_{1}) = \frac{3}{5} \cdot \frac{2}{4} = \frac{6}{20} = \frac{3}{10}$
Check by counting ordered pairs: there are $5 \times 4 = 20$ equally likely ordered draws, and $3 \times 2 = 6$ of them are (apple, orange). So the probability is $\frac{6}{20} = \frac{3}{10}$ , which is option (C).

KEY IDEA

Drawing in succession means drawing without replacement, so the events are dependent. For a specified order, multiply the probability of the first draw by the conditional probability of the second draw given the first: $P (A_{1} \cap O_{2}) = P (A_{1}) P (O_{2} ∣ A_{1})$ . Counting ordered pairs ( $6$ favourable out of $5 \times 4 = 20$ ) gives the same answer.

WATCH OUT FOR

Each wrong option matches a specific slip. (A) $\frac{3}{5}$ is $P (first is an apple)$ on its own. It is also the probability of one apple and one orange in either order, $2 \times \frac{3}{10}$ , so it is what you get if you ignore the order the question asks for. (D) $\frac{2}{5}$ is the probability of drawing an orange from the full box. Treating the draws as independent (with replacement) gives $\frac{3}{5} \cdot \frac{2}{5} = \frac{6}{25}$ , which is not an option. That is a sign the second draw has to use the $4$ fruits left in the box.

From the GATE 2026 IN paper. Every question of this paper · Download the full paper · All 33 Probability & statistics questions, with notes

GATE 2026 IN Q.36: solution and answer

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