| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Tree diagram with two-stage events |
| Difficulty | Moderate -0.8 This is a straightforward two-stage tree diagram problem requiring basic probability multiplication for part (i) and simple conditional probability (Bayes' theorem) for part (ii). Both parts use standard S1 techniques with clear given probabilities and no conceptual challenges beyond routine application of formulas. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(A \text{ Later}) = 0.5 \times 0.2 = 0.1\) | B1 [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(L \text{ given } I) = \frac{(0.2 \times 0.1)}{(0.5 \times 0.8 + 0.3 \times 0.6 + 0.2 \times 0.1)}\) | B1 | \(0.2 \times 0.1\) seen on its own as numerator or denominator of a fraction |
| Attempt at \(P(I)\) summing 2 or 3 two-factor products, seen anywhere | M1 | |
| \(= \frac{0.02}{0.6}\) | A1 | Correct unsimplified \(P(I)\) as numerator or denominator of a fraction |
| \(= 0.0333\ (1/30)\) | A1 [4] | Correct answer, accept 0.033 |
## Question 1:
**(i)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(A \text{ Later}) = 0.5 \times 0.2 = 0.1$ | B1 [1] | |
**(ii)**
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(L \text{ given } I) = \frac{(0.2 \times 0.1)}{(0.5 \times 0.8 + 0.3 \times 0.6 + 0.2 \times 0.1)}$ | B1 | $0.2 \times 0.1$ seen on its own as numerator or denominator of a fraction |
| Attempt at $P(I)$ summing 2 or 3 two-factor products, seen anywhere | M1 | |
| $= \frac{0.02}{0.6}$ | A1 | Correct unsimplified $P(I)$ as numerator or denominator of a fraction |
| $= 0.0333\ (1/30)$ | A1 [4] | Correct answer, accept 0.033 |
---1 Fabio drinks coffee each morning. He chooses Americano, Cappucino or Latte with probabilities 0.5, 0.3 and 0.2 respectively. If he chooses Americano he either drinks it immediately with probability 0.8 , or leaves it to drink later. If he chooses Cappucino he either drinks it immediately with probability 0.6 , or leaves it to drink later. If he chooses Latte he either drinks it immediately with probability 0.1 , or leaves it to drink later.\\
(i) Find the probability that Fabio chooses Americano and leaves it to drink later.\\
(ii) Fabio drinks his coffee immediately. Find the probability that he chose Latte.
\hfill \mbox{\textit{CAIE S1 2012 Q1 [5]}}