| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Exactly k successes in n trials |
| Difficulty | Standard +0.3 This is a straightforward conditional probability question requiring basic binomial probability calculations and application of P(A|B) = P(A∩B)/P(B). Part (i) involves computing probabilities for specific outcomes with given probabilities, and part (ii) is a standard conditional probability setup. The calculations are routine with no conceptual challenges beyond A-level S1 material. |
| Spec | 2.03a Mutually exclusive and independent events |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(SLL) = (0.3)(0.55)(0.55) = 0.09075 \left(\frac{363}{4000}\right)\) | M1 | \(P(SLL)\), \(P(SRR)\), \(P(SSL)\) or \(P(SSR)\) seen |
| \(P(SRR) = (0.3)(0.15)(0.15) = 0.00675 \left(\frac{27}{4000}\right)\) | A1 | Two correct options 0.09075 or 0.00675 can be unsimplified |
| Total \(= ^3C_1 \times P(SLL) + ^3C_1 \times P(SRR) = 0.27225 + 0.02025\) | M1 | Summing 6 prob options not all identical |
| \(\text{Prob} = 0.293\), accept \(0.2925 \left(\frac{117}{400}\right)\) | A1 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(SSS \mid \text{all same dir}^n) = \frac{P(SSS \text{ and same dir}^n)}{P(\text{same direction})}\) | B1 | \((0.3)^3\) oe seen on its own as num or denom of a fraction |
| M1 | Attempt at \(P(SSS + LLL + RRR)\) seen anywhere | |
| \(= \frac{0.3 \times 0.3 \times 0.3}{(0.15)^3 + (0.55)^3 + (0.3)^3}\) | A1 | \((0.15)^3 + (0.55)^3 + (0.3)^3\) oe seen as denom of a fraction |
| \(= 0.137 \left(\frac{108}{787}\right)\) | A1 | Correct answer |
## Question 6(i):
$P(SLL) = (0.3)(0.55)(0.55) = 0.09075 \left(\frac{363}{4000}\right)$ | M1 | $P(SLL)$, $P(SRR)$, $P(SSL)$ or $P(SSR)$ seen |
$P(SRR) = (0.3)(0.15)(0.15) = 0.00675 \left(\frac{27}{4000}\right)$ | A1 | Two correct options 0.09075 or 0.00675 can be unsimplified |
Total $= ^3C_1 \times P(SLL) + ^3C_1 \times P(SRR) = 0.27225 + 0.02025$ | M1 | Summing 6 prob options not all identical |
$\text{Prob} = 0.293$, accept $0.2925 \left(\frac{117}{400}\right)$ | A1 | Correct answer |
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## Question 6(ii):
$P(SSS \mid \text{all same dir}^n) = \frac{P(SSS \text{ and same dir}^n)}{P(\text{same direction})}$ | B1 | $(0.3)^3$ oe seen on its own as num or denom of a fraction |
| M1 | Attempt at $P(SSS + LLL + RRR)$ seen anywhere |
$= \frac{0.3 \times 0.3 \times 0.3}{(0.15)^3 + (0.55)^3 + (0.3)^3}$ | A1 | $(0.15)^3 + (0.55)^3 + (0.3)^3$ oe seen as denom of a fraction |
$= 0.137 \left(\frac{108}{787}\right)$ | A1 | Correct answer |6 Vehicles approaching a certain road junction from town $A$ can either turn left, turn right or go straight on. Over time it has been noted that of the vehicles approaching this particular junction from town $A$, $55 \%$ turn left, $15 \%$ turn right and $30 \%$ go straight on. The direction a vehicle takes at the junction is independent of the direction any other vehicle takes at the junction.\\
\begin{enumerate}[label=(\roman*)]
\item Find the probability that, of the next three vehicles approaching the junction from town $A$, one goes straight on and the other two either both turn left or both turn right.
\item Three vehicles approach the junction from town $A$. Given that all three drivers choose the same direction at the junction, find the probability that they all go straight on.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2018 Q6 [8]}}