Three coins are tossed, and the number of heads is recorded. Which of the following lists is a sample space for this experiment? Why do the other lists fail to qualify as a sample space?
- (i) {1, 2, 3}
- (ii) {0, 1, 2}
- (iii) {0, 1, 2, 3, 4}
- (iv) {0, 1, 2, 3}
Step-by-step solution
Idea: A sample space must include every possible outcome and nothing that cannot happen, each listed once. With 3 coins, the number of heads can be 0, 1, 2 or 3.
(i) {1, 2, 3}
- TTT gives 0 heads, but 0 is missing. Not a sample space: it leaves out a possible outcome.½ mark
(ii) {0, 1, 2}
- HHH gives 3 heads, but 3 is missing. Not a sample space: it leaves out a possible outcome.½ mark
(iii) {0, 1, 2, 3, 4}
- With only 3 coins you cannot get 4 heads. Not a sample space: it contains an outcome that cannot happen.½ mark
(iv) {0, 1, 2, 3}
- TTT → 0; HTT, THT, TTH → 1; HHT, HTH, THH → 2; HHH → 3. Every possible number of heads appears, once, and nothing else.1 mark
- This is the sample space. (Its outcomes are not equally likely: 1 and 2 heads each happen in 3 ways out of 8.)½ mark
Check: Number of heads ranges from 0 (all tails) to 3 (all heads), so exactly the four values 0, 1, 2, 3 are possible ✓.
Answer to write in the exam
(i)
TTT gives 0 heads; 0 is not listed.
∴ Not a sample space (a possible outcome is missing).
(ii)
HHH gives 3 heads; 3 is not listed.
∴ Not a sample space (a possible outcome is missing).
(iii)
Maximum number of heads with 3 coins = 3; 4 is impossible.
∴ Not a sample space (contains an impossible outcome).
(iv)
0: TTT; 1: HTT, THT, TTH; 2: HHT, HTH, THH; 3: HHH
∴ {0, 1, 2, 3} is the sample space.
Common mistakes that cost marks
- Forgetting that 0 heads is a possible result. All tails (TTT) is an outcome too.
- Choosing {0, 1, 2, 3, 4} “to be safe”. A sample space must not contain impossible outcomes.
- Thinking 0, 1, 2, 3 heads are equally likely (14 each). In fact P(0) = 18, P(1) = 38, P(2) = 38, P(3) = 18.
How this can come in the exam
Two coins are tossed and the number of tails is recorded. The sample space is
- {1, 2}
- {0, 1}
- {0, 1, 2}
- {0, 1, 2, 3}
Show answer
(C) {0, 1, 2}
With 2 coins you can get 0, 1 or 2 tails, and nothing else.
Assertion (A): {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} is a sample space for the sum of the numbers when two dice are thrown.
Reason (R): A sample space must contain every possible outcome, listed once, and no impossible outcome.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
Sums run from 2 to 12, all included once with nothing impossible; R is the rule that A satisfies.
Try one yourself
Four coins are tossed and the number of heads is recorded. Write the sample space.
Show answer
{0, 1, 2, 3, 4}.
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