| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Small sample binomial probability |
| Difficulty | Standard +0.3 Part (i) is straightforward algebra with probabilities summing to 1. Part (ii) is a simple counting problem with 3 spins. Part (iii) applies a standard normal approximation to binomial with continuity correction—a routine S1 technique. The question requires multiple steps but uses only standard methods with no novel insight, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(4p + p + 3p = 1\) so P(blue) = 1/8 AG | B1 | [1] Must show something |
| (ii) P(R) = 1/2, P(B) = 1/8, P(G) = 3/8; P(all different) = \(\frac{1}{2} \times \frac{1}{8} \times \frac{3}{8} \times 3!\) = 9/64 (0.141) | M1, M1, A1 | [3] Multiplying P(R, B, G) together; Mult by 3!; Correct answer |
**(i)** $4p + p + 3p = 1$ so P(blue) = 1/8 AG | B1 | [1] Must show something
**(ii)** P(R) = 1/2, P(B) = 1/8, P(G) = 3/8; P(all different) = $\frac{1}{2} \times \frac{1}{8} \times \frac{3}{8} \times 3!$ = 9/64 (0.141) | M1, M1, A1 | [3] Multiplying P(R, B, G) together; Mult by 3!; Correct answer5 A triangular spinner has one red side, one blue side and one green side. The red side is weighted so that the spinner is four times more likely to land on the red side than on the blue side. The green side is weighted so that the spinner is three times more likely to land on the green side than on the blue side.\\
(i) Show that the probability that the spinner lands on the blue side is $\frac { 1 } { 8 }$.\\
(ii) The spinner is spun 3 times. Find the probability that it lands on a different coloured side each time.\\
(iii) The spinner is spun 136 times. Use a suitable approximation to find the probability that it lands on the blue side fewer than 20 times.
\hfill \mbox{\textit{CAIE S1 2011 Q5 [9]}}