| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Bayes with complementary outcome |
| Difficulty | Moderate -0.3 This is a straightforward application of conditional probability and Bayes' theorem with clearly defined probabilities and outcomes. Part (i) requires simple multiplication of probabilities, while part (ii) applies Bayes' theorem with complementary events. The structure is standard textbook material with no conceptual challenges beyond basic probability rules. |
| Spec | 2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(P(C \cap < 50) = 0.35 \times 0.2 = 0.07\) | B1 [1] | |
| (ii) \(P(C | < 50) = \frac{P(C \cap < 50)}{P(< 50)}\) | M1 [4] |
| \(= \frac{0.35 \times 0.2}{0.25 \times 0.3 + 0.35 \times 0.2 + 0.4(<1)}\) | A1 | 0.545 (unsimplified) seen as num or denom of a fraction |
| \(= \frac{0.07}{0.545}\) | M1 | Attempt at P(C < 50) as 2-factor prod only seen as num or denom of a fraction |
| \(= 0.128 \left(\frac{14}{109}\right)\) | A1 | Correct answer |
**(i)** $P(C \cap < 50) = 0.35 \times 0.2 = 0.07$ | B1 [1] |
**(ii)** $P(C | < 50) = \frac{P(C \cap < 50)}{P(< 50)}$ | M1 [4] | Summing three 2-factor products seen anywhere (can omit the 1)
$= \frac{0.35 \times 0.2}{0.25 \times 0.3 + 0.35 \times 0.2 + 0.4(<1)}$ | A1 | 0.545 (unsimplified) seen as num or denom of a fraction
$= \frac{0.07}{0.545}$ | M1 | Attempt at P(C < 50) as 2-factor prod only seen as num or denom of a fraction
$= 0.128 \left(\frac{14}{109}\right)$ | A1 | Correct answer2 On Saturday afternoons Mohit goes shopping with probability 0.25, or goes to the cinema with probability 0.35 or stays at home. If he goes shopping the probability that he spends more than $\$ 50$ is 0.7 . If he goes to the cinema the probability that he spends more than $\$ 50$ is 0.8 . If he stays at home he spends $\$ 10$ on a pizza.\\
(i) Find the probability that Mohit will go to the cinema and spend less than $\$ 50$.\\
(ii) Given that he spends less than $\$ 50$, find the probability that he went to the cinema.
\hfill \mbox{\textit{CAIE S1 2013 Q2 [5]}}