GATE 2020 IN Q.16: solution and answer

GATE 2020 · INQ.16 · 1 mark · NAT

A player throws a ball at a basket kept at a distance. The probability that the ball falls into the basket in a single attempt is 0.1. The player attempts to throw the ball twice. Considering each attempt to be independent, the probability that this player puts the ball into the basket only in the second attempt (rounded off to two decimal places) is ____

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Answer and solution

ANSWER

The probability that the player puts the ball into the basket only in the second attempt is $0.9 \times 0.1 = 0.09$ .

Official answer: 0.09Source: Official answer key, GATE 2020 IN (IIT Delhi)

Let $S$ stand for a throw that goes in and $F$ for a miss. For a single attempt $P (S) = 0.1$ , so by the complement rule $P (F) = 1 - 0.1 = 0.9$ .
"Only in the second attempt" describes exactly one outcome of the two throws: the first throw misses and the second goes in. That is the event $F_{1} \cap S_{2}$ . The outcomes $S_{1} S_{2}$ and $S_{1} F_{2}$ are excluded because the ball went in on the first throw, and $F_{1} F_{2}$ is excluded because it never went in.
The attempts are independent, so the multiplication rule gives $P (F_{1} \cap S_{2}) = P (F_{1}) P (S_{2}) = 0.9 \times 0.1 = 0.09$
$0.09$ already has two decimal places, so no rounding is needed and the answer is $0.09$ . This is also the geometric-distribution probability that the first success comes on trial $k = 2$ : $(1 - p)^{k - 1} p = 0.9 \times 0.1$ .

KEY IDEA

Read "only in the second attempt" as one specific sequence: a miss on the first throw, then a basket on the second. Independence lets you multiply the two probabilities: $P (F_{1}) P (S_{2}) = (1 - p) p$ . This is the geometric probability that the first success comes on trial 2.

WATCH OUT FOR

Answering $0.1$ by ignoring the requirement that the first throw misses. Another error is answering $0.18$ , which is $P (exactly one success in two throws) = 2 \times 0.9 \times 0.1$ and wrongly includes the case where the ball goes in on the first throw only. A third is answering $0.01$ , which is the probability of scoring on both throws.

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GATE 2020 IN Q.16: solution and answer

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