GATE 2022 IN GA Q.7: solution and answer

GATE 2022 · INGA Q.7 · 2 marks · MCQ

A game consists of spinning an arrow around a stationary disk as shown below. When the arrow comes to rest, there are eight equally likely outcomes. It could come to rest in any one of the sectors numbered 1, 2, 3, 4, 5, 6, 7 or 8 as shown.

Two such disks are used in a game where their arrows are independently spun.

What is the probability that the sum of the numbers on the resulting sectors upon spinning the two disks is equal to 8 after the arrows come to rest?

GATE 2022 IN GA question 7: Two identical circular disks drawn side by side. — Figure from the GATE 2022 IN paper. It is part of the question.

(A)
$\frac{1}{16}$
(B)
$\frac{5}{64}$
(C)
$\frac{3}{32}$
(D)
$\frac{7}{64}$

Answer and solution

ANSWER

The probability that the two numbers add up to 8 is $\frac{7}{64}$ , which is option (D).

Official answer: DSource: Official answer key, GATE 2022 IN (IIT Kharagpur)

Sample space: each disk has 8 equally likely outcomes. The two spins are independent, so there are $8 \times 8 = 64$ equally likely ordered pairs $(a, b)$ , where $a$ is the number on the first disk and $b$ is the number on the second.
Favourable outcomes: we need $a + b = 8$ with $1 \leq a, b \leq 8$ . For each $a$ from 1 to 7 there is exactly one partner $b = 8 - a$ , and it lies between 1 and 7, so it is on the disk. The value $a = 8$ would need $b = 0$ , which is not on the disk.
So the favourable pairs are $(1, 7), (2, 6), (3, 5), (4, 4), (5, 3), (6, 2), (7, 1)$ , which makes 7 ordered pairs. $(1, 7)$ and $(7, 1)$ count separately because the two disks are distinct. $(4, 4)$ counts once.
Classical probability, favourable over total: $P (sum = 8) = \frac{7}{64}$
This matches option (D).

KEY IDEA

Two independent spins of an 8-sector disk give $8 \times 8 = 64$ equally likely ordered outcomes. Count the ordered pairs $(a, b)$ with $a + b = 8$ and divide by 64. For a target sum $s$ with $2 \leq s \leq 9$ , there are $s - 1$ such pairs, so here there are 7.

WATCH OUT FOR

Counting unordered pairs, so that ${1, 7}, {2, 6}, {3, 5}, {4, 4}$ gives 4 outcomes, then dividing by 64 or by 36. The 64 outcomes are ordered pairs, so the favourable outcomes must be ordered too. A second slip is to include $(8, 0)$ or $(0, 8)$ , but 0 is not a sector on the disk.

From the GATE 2022 IN paper. Every question of this paper · Download the full paper · All 7 Statistics & probability questions, with notes

GATE 2022 IN GA Q.7: solution and answer

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