| 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.\\
(i) 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.\\
(ii) 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.\\
\hfill \mbox{\textit{CAIE S1 2018 Q6 [8]}}