| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Direct binomial probability calculation |
| Difficulty | Moderate -0.8 This is a straightforward S1 binomial probability question requiring basic application of the binomial formula with clearly stated probabilities (15% choose Music for boys, calculating P(X<3) for n=6). The calculation involves standard binomial probability with no conceptual challenges—simpler than typical A-level questions as it's pure recall and arithmetic. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) P(0, 1, 2) = \((0.85)^3 + (0.15)(0.85)^2C_1 + (0.15)^2(0.85)^1C_2\) = 0.953 | B1, M1, A1 | [3] 0.15 and 0.85 seen; Any binomial expression Σpowers = 6, Σ p = 1; Correct answer |
| (ii) P(D) = 0.6 × 0.1 + 0.4 × 0.55 = 0.28; P(B | D) = \(\frac{P(B \cap D)}{P(D)}\) = 0.06/0.28 = 0.2143 | M1, A1, M1, √A1 |
| P(> 1) = 1 − P(0) = \(1 − (0.7857)^5\) = 1 − 0.7078 = 0.70 | M1, A1 | [6] Binomial expression 1−P(0) or 1−P(0, 1) Σ p = 1; Correct answer accept 0.700 |
**(i)** P(0, 1, 2) = $(0.85)^3 + (0.15)(0.85)^2C_1 + (0.15)^2(0.85)^1C_2$ = 0.953 | B1, M1, A1 | [3] 0.15 and 0.85 seen; Any binomial expression Σpowers = 6, Σ p = 1; Correct answer
**(ii)** P(D) = 0.6 × 0.1 + 0.4 × 0.55 = 0.28; P(B|D) = $\frac{P(B \cap D)}{P(D)}$ = 0.06/0.28 = 0.2143 | M1, A1, M1, √A1 | Attempt to find P(D); 0.28 seen; Using cond prob formula to find P(B|D); Correct unsimplified answer
P(> 1) = 1 − P(0) = $1 − (0.7857)^5$ = 1 − 0.7078 = 0.70 | M1, A1 | [6] Binomial expression 1−P(0) or 1−P(0, 1) Σ p = 1; Correct answer accept 0.7006 There are a large number of students in Luttley College. $60 \%$ of the students are boys. Students can choose exactly one of Games, Drama or Music on Friday afternoons. It is found that $75 \%$ of the boys choose Games, $10 \%$ of the boys choose Drama and the remainder of the boys choose Music. Of the girls, $30 \%$ choose Games, $55 \%$ choose Drama and the remainder choose Music.\\
(i) 6 boys are chosen at random. Find the probability that fewer than 3 of them choose Music.\\
(ii) 5 Drama students are chosen at random. Find the probability that at least 1 of them is a boy.\\
(i) In a certain country, the daily minimum temperature, in ${ } ^ { \circ } \mathrm { C }$, in winter has the distribution $\mathrm { N } ( 8,24 )$. Find the probability that a randomly chosen winter day in this country has a minimum temperature between $7 ^ { \circ } \mathrm { C }$ and $12 ^ { \circ } \mathrm { C }$.
The daily minimum temperature, in ${ } ^ { \circ } \mathrm { C }$, in another country in winter has a normal distribution with mean $\mu$ and standard deviation $2 \mu$.\\
(ii) Find the proportion of winter days on which the minimum temperature is below zero.\\
(iii) 70 winter days are chosen at random. Find how many of these would be expected to have a minimum temperature which is more than three times the mean.\\
(iv) The probability of the minimum temperature being above $6 ^ { \circ } \mathrm { C }$ on any winter day is 0.0735 . Find the value of $\mu$.
\hfill \mbox{\textit{CAIE S1 2011 Q6 [9]}}