Question
Three coins are tossed simultaneously. Find the probability of getting exactly 2 heads.
(CBSE 2024)
Solution — Step by Step
Each coin has 2 outcomes (H or T). For 3 coins, total outcomes .
The complete sample space: {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
We need exactly 2 heads (and therefore exactly 1 tail). Let’s pick them out:
- HHT (tail in 3rd position)
- HTH (tail in 2nd position)
- THH (tail in 1st position)
Favourable outcomes = 3
The answer is (or 0.375).
Why This Works
The reason there are exactly 3 ways to get 2 heads from 3 coins is a counting argument: we’re choosing which 2 out of 3 positions will show heads. That’s .
This connects to the binomial coefficient — and for higher classes, the formula becomes . But at the Class 10 level, listing the sample space is the expected method.
Each of the 8 outcomes in the sample space is equally likely (probability each), because we assume fair coins. The probability of any event is simply the count of favourable outcomes divided by 8.
Alternative Method — Tree Diagram
Draw a tree with 3 levels (one per coin). Each branch splits into H and T. Count the paths that pass through exactly 2 H’s. You’ll find 3 such paths, confirming .
For CBSE, always write out the full sample space when the total outcomes are small (8 or fewer). The marking scheme awards 1 mark for writing the sample space correctly. Skipping it and jumping to the answer can cost you marks even if the final answer is right.
Common Mistake
Students sometimes write only 4 outcomes: {0 heads, 1 head, 2 heads, 3 heads} and say . This is wrong because these 4 events are NOT equally likely. “2 heads” has 3 ways while “3 heads” has only 1 way. You must count at the level of individual coin outcomes (HHT, HTH, etc.) where each outcome is equally probable.