| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Multi-stage game or match outcomes |
| Difficulty | Standard +0.3 This is a standard two-stage conditional probability problem requiring a tree diagram and application of Bayes' theorem. While it involves multiple probabilities and careful bookkeeping, the techniques are routine for A-level statistics: constructing a tree diagram from given conditional probabilities and using P(A|B) = P(A∩B)/P(B). The arithmetic is straightforward and the problem structure is typical of textbook exercises. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Tree diagram with correct shape: 3 branches (\(W: 3/5\), \(D: 1/5\), \(L: 1/5\)) then 3 branches each. Second level: from \(W\): \((W: 7/10, D: 2/10, L: 1/10)\); from \(D\): \((W: 1/3, D: 1/3, L: 1/3)\); from \(L\): \((W: 3/10, D: 1/20, L: 13/20)\) | M1 | Correct shape i.e. 3 branches then 3 by 3 branches, labelled and clear annotation. Condone omission of lines for first match result providing the probabilities are there. |
| All correct probabilities | A1 | All correct probs with fully correct shape and probs either fractions or decimals not 1.5/5 etc. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(L_1 \text{ given } W_2) = \frac{P(L_1 \cap W_2)}{P(W_2)}\) | M1 | Attempt at \(P(L_1 \cap W_2)\) as a two-factor product only as numerator or denominator of a fraction |
| \(= \frac{1/5 \times 3/10}{3/5 \times 7/10 + 1/5 \times 1/3 + 1/5 \times 3/10}\) | M1 | Attempt at \(P(W_2)\) as sum of appropriate 3 two-factor products OE seen anywhere |
| A1 | Unsimplified correct \(P(W_2)\) numerator or denominator of a fraction | |
| \(= \frac{3/50}{41/75} = 9/82 \ (0.110)\) | A1 |
## Question 3:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Tree diagram with correct shape: 3 branches ($W: 3/5$, $D: 1/5$, $L: 1/5$) then 3 branches each. Second level: from $W$: $(W: 7/10, D: 2/10, L: 1/10)$; from $D$: $(W: 1/3, D: 1/3, L: 1/3)$; from $L$: $(W: 3/10, D: 1/20, L: 13/20)$ | M1 | Correct shape i.e. 3 branches then 3 by 3 branches, labelled and clear annotation. Condone omission of lines for first match result providing the probabilities are there. |
| All correct probabilities | A1 | All correct probs with fully correct shape and probs either fractions or decimals not 1.5/5 etc. |
## Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(L_1 \text{ given } W_2) = \frac{P(L_1 \cap W_2)}{P(W_2)}$ | M1 | Attempt at $P(L_1 \cap W_2)$ as a two-factor product only as numerator or denominator of a fraction |
| $= \frac{1/5 \times 3/10}{3/5 \times 7/10 + 1/5 \times 1/3 + 1/5 \times 3/10}$ | M1 | Attempt at $P(W_2)$ as sum of appropriate 3 two-factor products OE seen anywhere |
| | A1 | Unsimplified correct $P(W_2)$ numerator or denominator of a fraction |
| $= \frac{3/50}{41/75} = 9/82 \ (0.110)$ | A1 | |
---3 Redbury United soccer team play a match every week. Each match can be won, drawn or lost. At the beginning of the soccer season the probability that Redbury United win their first match is $\frac { 3 } { 5 }$, with equal probabilities of losing or drawing. If they win the first match, the probability that they win the second match is $\frac { 7 } { 10 }$ and the probability that they lose the second match is $\frac { 1 } { 10 }$. If they draw the first match they are equally likely to win, draw or lose the second match. If they lose the first match, the probability that they win the second match is $\frac { 3 } { 10 }$ and the probability that they draw the second match is $\frac { 1 } { 20 }$.\\
(i) Draw a fully labelled tree diagram to represent the first two matches played by Redbury United in the soccer season.\\
(ii) Given that Redbury United win the second match, find the probability that they lose the first match.\\
\hfill \mbox{\textit{CAIE S1 2017 Q3 [6]}}