| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | General probability threshold |
| Difficulty | Standard +0.3 Part (i) requires recognizing that 'medium or large' means p=0.85, then calculating P(X<12) for a binomial distribution—straightforward but involves multiple probability calculations. Part (ii) requires setting up (0.85)^n ≥ 0.1 and solving using logarithms, which is a standard technique but requires careful algebraic manipulation. Both parts are routine applications of binomial distribution with no novel insight required, slightly above average due to the two-part structure and logarithm work. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(< 12) = 1 - P(12, 13, 14) = 1 - [(0.85)^{12}(0.15)^2 {}_14C_{12} + (0.85)^{13}(0.15)^1 + (0.85)^{14}] = 1 - 0.6479 = 0.352\) | B1 | \((p = )0.85\) oe seen anywhere |
| M1 | Summing 2 or 3 consistent bin probs, any \(p < 1\), \(n = 14\) (or summing 12 or 13 consistent bin probs) | |
| A1 3 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(n = 14\) | M1 | Eqn or inequality in 0.85(or 0.15), \(n\), 0.1, \(n\) as a power |
| M1 | Attempt to solve (can be implied) if \(n\) a power | |
| A1 3 | Correct answer – must be equals, not approx. MR allowed for 0.01, M1M1A0 max. |
(i) $(p = )0.85$
$P(< 12) = 1 - P(12, 13, 14) = 1 - [(0.85)^{12}(0.15)^2 {}_14C_{12} + (0.85)^{13}(0.15)^1 + (0.85)^{14}] = 1 - 0.6479 = 0.352$ | B1 | $(p = )0.85$ oe seen anywhere
| M1 | Summing 2 or 3 consistent bin probs, any $p < 1$, $n = 14$ (or summing 12 or 13 consistent bin probs)
| A1 3 | Correct answer
(ii) $(0.85)^n \geq 0.1$
$n \leq 14.2$
$n = 14$ | M1 | Eqn or inequality in 0.85(or 0.15), $n$, 0.1, $n$ as a power
| M1 | Attempt to solve (can be implied) if $n$ a power
| A1 3 | Correct answer – must be equals, not approx. MR allowed for 0.01, M1M1A0 max.3 In a large consignment of mangoes, 15\% of mangoes are classified as small, 70\% as medium and $15 \%$ as large.\\
(i) Yue-chen picks 14 mangoes at random. Find the probability that fewer than 12 of them are medium or large.\\
(ii) Yue-chen picks $n$ mangoes at random. The probability that none of these $n$ mangoes is small is at least 0.1 . Find the largest possible value of $n$.
\hfill \mbox{\textit{CAIE S1 2013 Q3 [6]}}