There are two identical dice with a single letter on each of the faces. The following six letters: Q, R, S, T, U, and V, one on each of the faces. Any of the six outcomes are equally likely.
The two dice are thrown once independently at random.
What is the probability that the outcomes on the dice were composed only of any combination of the following possible outcomes: Q, U and V?
ANSWER
The probability is , option (A). Each die shows Q, U or V with probability , the dice are independent, and .
Official answer: ASource: Official answer key, GATE 2022 EE (IIT Kharagpur)
Read the event correctly: "composed only of any combination of Q, U and V" means each die shows one of Q, U or V. Repeats such as (Q, Q) count, and the two dice do not need to show different letters.
Sample space: each die has 6 equally likely faces and the throws are independent. The dice are identical, but they are still two separate objects, so there are equally likely ordered outcomes.
One die: 3 of its 6 faces (Q, U, V) are favourable, so .
The dice are independent, so multiply: .
Check by counting: the favourable ordered pairs number (QQ, QU, QV, UQ, UU, UV, VQ, VU, VV), so . This is option (A).
KEY IDEA
"Only Q, U and V" asks that every die shows a letter from that set. For independent dice, multiply the single-die probability: . "Identical" dice are still two separate dice, so count the 36 ordered outcomes, which are equally likely.
WATCH OUT FOR
Counting only pairs with two different letters from {Q, U, V} (QU, QV, UV and their reverses, 6 outcomes) gives , which is trap option (C). The wording "any combination" allows repeats like QQ. A second trap is treating the "identical" dice as giving 21 unordered outcomes: those outcomes are not equally likely, and the count gives , which is not an option. Option (B), , is the complement: the probability that at least one die shows R, S or T.
From the GATE 2022 EE paper. Every question of this paper · Download the full paper · All 7 Statistics & probability questions, with notes
