| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Conditional probability in selection |
| Difficulty | Standard +0.3 This is a straightforward conditional probability question using combinations for card selection. Part (i) requires basic counting of favorable outcomes, part (ii) involves systematic enumeration of all possible sums, and parts (iii-iv) apply standard definitions of independence and exclusivity. The calculations are routine for S1 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.03a Mutually exclusive and independent events2.04a Discrete probability distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Probs \(\frac{(4/10 \times 6/9 \times 5/8) \times 3C1}{= 360/720 = 1/2}\) AG | M1, M1, A1 [3] | Summing three 3-factor options oe; 10 × 9 × 8 seen in denom; Correct answer |
| OR \(\frac{_6C_2 \times _4C_1}{_{10}C_3} = \frac{1}{2}\) AG | M1, M1, A1 | One of 6C2 or 4C1 seen in num; 10C3 in denom; Correct answer |
| (ii) | B1 [4] | 9, 10, 11, 12 only seen |
| sum | 9 | 10 |
| Prob | 24/720 | 216/720 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(4, 4, 4) = 6/10 \times 5/9 \times 4/8 = 120/720(1/6)\) | B1, B1, B1 | One correct prob other than P(11), with or without replacement; Another correct prob; \(\Sigma\) all 4 probs \(= 1\) |
| (iii) \(P(R) = 0.5\) \(P(S) = 0.4\) \(P(R \cap S) = 120/720\) | B1 [3] | \(P(R \cap S) = 120/720\) (1/6); Numerical attempt to compare \(P(R\) and \(S)\) with \(P(R) \times P(S)\) provided \(P(R \cap S) \neq 1/5\) |
| Answer | Marks | Guidance |
|---|---|---|
| Not indep | A1 ft | Correct conclusion ft wrong \(P(R \cap S) \neq 1/5\), \(P(S)\) correct |
| Answer | Marks | Guidance |
|---|---|---|
| Not exclusive \(\Sigma xf/\Sigma f\) | B1 ft [1] | Correct answer following correct reasoning ft wrong non zero \(P(R \cap S)\) |
**(i)** options (3, 4, 4) or (4, 3, 4) or (4, 4, 3)
Probs $\frac{(4/10 \times 6/9 \times 5/8) \times 3C1}{= 360/720 = 1/2}$ AG | M1, M1, A1 [3] | Summing three 3-factor options oe; 10 × 9 × 8 seen in denom; Correct answer
OR $\frac{_6C_2 \times _4C_1}{_{10}C_3} = \frac{1}{2}$ AG | M1, M1, A1 | One of 6C2 or 4C1 seen in num; 10C3 in denom; Correct answer
**(ii)** | B1 [4] | 9, 10, 11, 12 only seen
| sum | 9 | 10 | 11 | 12 |
|---|---|---|---|---|
| Prob | 24/720 | 216/720 | 360/720 | 120/720 |
$P(3, 3, 3) = 4/10 \times 3/9 \times 2/8 = 24/720$ (1/30)
$P(3, 3, 4) = 4/10 \times 3/9 \times 6/8 \times 3C1 = 216/720$ (3/10)
$P(4, 4, 4) = 6/10 \times 5/9 \times 4/8 = 120/720(1/6)$ | B1, B1, B1 | One correct prob other than P(11), with or without replacement; Another correct prob; $\Sigma$ all 4 probs $= 1$
**(iii)** $P(R) = 0.5$ $P(S) = 0.4$ $P(R \cap S) = 120/720$ | B1 [3] | $P(R \cap S) = 120/720$ (1/6); Numerical attempt to compare $P(R$ and $S)$ with $P(R) \times P(S)$ provided $P(R \cap S) \neq 1/5$
$P(R \cap S) = 120/720 \neq P(R) \times P(S)$
Not indep | A1 ft | Correct conclusion ft wrong $P(R \cap S) \neq 1/5$, $P(S)$ correct
**(iv)** $P(R \cap S) \neq 0$ or there is an overlap between $R$ and $S$ (34,4)
Not exclusive $\Sigma xf/\Sigma f$ | B1 ft [1] | Correct answer following correct reasoning ft wrong non zero $P(R \cap S)$7 Rory has 10 cards. Four of the cards have a 3 printed on them and six of the cards have a 4 printed on them. He takes three cards at random, without replacement, and adds up the numbers on the cards.\\
(i) Show that P (the sum of the numbers on the three cards is $11 ) = \frac { 1 } { 2 }$.\\
(ii) Draw up a probability distribution table for the sum of the numbers on the three cards.
Event $R$ is 'the sum of the numbers on the three cards is 11 '. Event $S$ is 'the number on the first card taken is a $3 ^ { \prime }$.\\
(iii) Determine whether events $R$ and $S$ are independent. Justify your answer.\\
(iv) Determine whether events $R$ and $S$ are exclusive. Justify your answer.
\hfill \mbox{\textit{CAIE S1 2013 Q7 [11]}}