Worked solution
Separate the two-card stopping cases from the squared third-card scores
Four distinct cards are numbered 1,2,3,4. A player randomly picks a pair without replacement. If their sum is odd, that sum is the score. If the sum is even, the player randomly picks a third card from the remaining cards and scores the square of its number.
(i) Find the probability of score 4.
(ii) Find the probability distribution of score S, and hence its expectation.
(iii) Find the probability of a score below 5, given three cards are drawn.
[2017/YJC/II/7]
CJC · Discussion · Question 2
Answer
(i) ; (ii) distribution above, ; (iii) .
(i) Score 4 requires the third card to be 2
The first pair must be{1,3}, in either order, followed by 2. Thus .
(ii) Retain the full source distribution
| s | |
|---|---|
| 1 | |
| 3 | |
| 4 | |
| 5 | |
| 7 | |
| 9 | |
| 16 |
The odd-score pairs{1,2},{1,4},{2,3},{3,4} stop after two draws; score 5 collects two such pairs. The even first pairs{1,3} or{2,4} lead to squared third-card scores 1,4,9,16. Each of those squared-score cases has probability 1/12.
(ii) Use every source weighted term
.
(iii) Condition on the third draw happening
A third draw occurs exactly when the first pair is{1,3} or{2,4}, with probability . Among these paths, score below 5 means first pair{1,3} then 2 (score 4), or first pair{2,4} then 1 (score 1). Their total probability is . The source conditional quotient is therefore .
Common mistakes
Score 3 is below 5 but is impossible after a third draw; the conditioning event changes which scores count.
Solution notes
All three asks, complete seven-value source distribution, every weighted expectation term and full conditional numerator/denominator retained. Pair/third-card descriptions clarify the source’s compact card-sequence notation. No marks printed.