JEE Main 2021 · 27 August · evening · Single correct · +4 / −1
Each of the persons A and B independently tosses three fair coins. The probability that both of them get the same number of heads is :
- (a) 1/8
- (b) 5/8
- (c) 5/16
- (d) 1
Answer and solution
Answer (c) 5/16
Solution
Maths sir · 3 steps
Split by the number of heads, find the chance both land on that number, then add the four cases.
- One person first. Three fair coins give 2³ = 8 equally likely outcomes. The chance of exactly k heads is C(3, k)/8, which gives 1/8, 3/8, 3/8, 1/8 for k = 0, 1, 2, 3.
- Both match at k. A and B toss independently, so the chance that both get exactly k heads is P(k) × P(k) = P(k)². The cases k = 0, 1, 2, 3 cannot happen together, so their probabilities add.
Heads k P(A gets k) P(both get k) 0 1/8 1/64 1 3/8 9/64 2 3/8 9/64 3 1/8 1/64 P(same) = (1 + 9 + 9 + 1)/64 = 20/64 = 5/16
- So. The probability that A and B get the same number of heads is 5/16, which is option (c).
The idea to keep. For two independent people to match, add P(k)² over every possible value k. Matching counts is easier than matching exact sequences, so the answer is bigger than 1/8.
Answer from the NTA final answer key. Solution by PrepWiser.