| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Finding unknown probability from total probability |
| Difficulty | Moderate -0.8 This is a straightforward application of the law of total probability and Bayes' theorem with clearly defined probabilities. Part (i) requires simple multiplication (0.25 × p = 0.075), and part (ii) is a standard conditional probability calculation using given values. The question involves routine manipulation of probability formulas with no conceptual challenges or novel problem-solving required. |
| Spec | 2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(0.25p = 0.075\) so \(p = 0.075/0.25 = 0.3\) | B1 [1] | Answer given, must show some working |
| (ii) \(P(2 | M) = \frac{P(2 \text{ and } M)}{P(M)}\) | M1 |
| \(= \frac{0.45 \times 0.85}{0.3 \times 0.1 + 0.45 \times 0.85 + 0.25 \times 0.3}\) | B1 | correct numerator of a fraction |
| \(= \frac{0.3825}{0.4875}\) | A1 | correct unsimplified denom |
| \(= 0.785\) | A1 [4] | correct answer |
**(i)** $0.25p = 0.075$ so $p = 0.075/0.25 = 0.3$ | B1 [1] | Answer given, must show some working
**(ii)** $P(2|M) = \frac{P(2 \text{ and } M)}{P(M)}$ | M1 | attempt at cond prob with single prod in num and $\sum$ three 2-factor o.e prods in denom
$= \frac{0.45 \times 0.85}{0.3 \times 0.1 + 0.45 \times 0.85 + 0.25 \times 0.3}$ | B1 | correct numerator of a fraction
$= \frac{0.3825}{0.4875}$ | A1 | correct unsimplified denom
$= 0.785$ | A1 [4] | correct answer2 Maria has 3 pre-set stations on her radio. When she switches her radio on, there is a probability of 0.3 that it will be set to station 1, a probability of 0.45 that it will be set to station 2 and a probability of 0.25 that it will be set to station 3 . On station 1 the probability that the presenter is male is 0.1 , on station 2 the probability that the presenter is male is 0.85 and on station 3 the probability that the presenter is male is $p$. When Maria switches on the radio, the probability that it is set to station 3 and the presenter is male is 0.075 .\\
(i) Show that the value of $p$ is 0.3 .\\
(ii) Given that Maria switches on and hears a male presenter, find the probability that the radio was set to station 2.
\hfill \mbox{\textit{CAIE S1 2012 Q2 [5]}}