| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Given conditional, find joint or marginal |
| Difficulty | Moderate -0.8 This is a straightforward conditional probability question using the standard tree diagram or two-way table approach. Part (i) requires simple probability calculations using complements and the law of total probability, while part (ii) applies Bayes' theorem in a direct manner with all necessary probabilities given explicitly. The question involves routine manipulation of given percentages with no conceptual challenges beyond basic conditional probability definitions. |
| Spec | 2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(S) = 0.65 \times 0.6 + 0.35 \times 0.75\) | M1 | Summing two 2-factor probs or \(1 -\) (sum of two 2-factor probs) |
| \(= 0.653\ (261/400)\) | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(Std \mid L) = \dfrac{P(Std \cap L)}{P(L)} = \dfrac{0.35 \times 0.25}{1 - 0.6525} = 0.0875/0.3475\) | M1 | \(P(Std)' \times P(L/Std)\) as numerator of fraction. Could be from tree diagram in 3(i) |
| M1 | Denominator \((1 - \text{their (i)})\) or their (i); or \(0.65 \times 0.4 + 0.35 \times 0.25 = 0.26 + 0.0875\); or \(P(L)\) from their tree diagram | |
| \(= 0.252\ (35/139)\) | A1 | |
| Total: 3 |
# Question 3(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(S) = 0.65 \times 0.6 + 0.35 \times 0.75$ | M1 | Summing two 2-factor probs or $1 -$ (sum of two 2-factor probs) |
| $= 0.653\ (261/400)$ | A1 | |
| **Total: 2** | | |
# Question 3(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(Std \mid L) = \dfrac{P(Std \cap L)}{P(L)} = \dfrac{0.35 \times 0.25}{1 - 0.6525} = 0.0875/0.3475$ | M1 | $P(Std)' \times P(L/Std)$ as numerator of fraction. Could be from tree diagram in 3(i) |
| | M1 | Denominator $(1 - \text{their (i)})$ or their (i); **or** $0.65 \times 0.4 + 0.35 \times 0.25 = 0.26 + 0.0875$; **or** $P(L)$ from their tree diagram |
| $= 0.252\ (35/139)$ | A1 | |
| **Total: 3** | | |3 A shop sells two makes of coffee, Café Premium and Café Standard. Both coffees come in two sizes, large jars and small jars. Of the jars on sale, $65 \%$ are Café Premium and $35 \%$ are Café Standard. Of the Café Premium, 40\% of the jars are large and of the Café Standard, 25\% of the jars are large. A jar is chosen at random.\\
(i) Find the probability that the jar is small.\\
(ii) Find the probability that the jar is Café Standard given that it is large.\\
\hfill \mbox{\textit{CAIE S1 2017 Q3 [5]}}