| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Combined probability with other distributions |
| Difficulty | Standard +0.3 This question requires finding a probability from the distribution table (straightforward: 4p = 0.6, so p = 0.15), then applying binomial probability for 'at most 1 success' in 3 trials. It's a standard two-step problem combining basic probability with routine binomial calculation, slightly easier than average due to small numbers and clear structure. |
| Spec | 2.04a Discrete probability distributions2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 |
| \(\mathrm { P } ( X = x )\) | \(p\) | \(p\) | \(p\) | \(p\) | 0.2 | 0.2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(\text{throwing a 4}) = (1 - 0.4)/4 = 0.15\) | M1, A1 | Sensible attempt to find \(P(1)\); Correct answer |
| \(P(\text{at most } 1) = P(0,1)\) or \(1 - P(2,3) = (0.85)^3 + {}^3C_1(0.15)(0.85)^2\) | M1, M1 | A binomial term with \({}^3C_n\) or any \(p\); Binomial expression with \({}^3C_n\) \(P(0,1)\) or \(1-P(2,3)\); \(p = 0.15\) or \(0.85\) |
| \(= 0.939\) | A1 [5] |
## Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{throwing a 4}) = (1 - 0.4)/4 = 0.15$ | M1, A1 | Sensible attempt to find $P(1)$; Correct answer |
| $P(\text{at most } 1) = P(0,1)$ or $1 - P(2,3) = (0.85)^3 + {}^3C_1(0.15)(0.85)^2$ | M1, M1 | A binomial term with ${}^3C_n$ or any $p$; Binomial expression with ${}^3C_n$ $P(0,1)$ or $1-P(2,3)$; $p = 0.15$ or $0.85$ |
| $= 0.939$ | A1 [5] | |
---2 The faces of a biased die are numbered $1,2,3,4,5$ and 6 . The random variable $X$ is the score when the die is thrown. The following is the probability distribution table for $X$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
$x$ & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
$\mathrm { P } ( X = x )$ & $p$ & $p$ & $p$ & $p$ & 0.2 & 0.2 \\
\hline
\end{tabular}
\end{center}
The die is thrown 3 times. Find the probability that the score is 4 on not more than 1 of the 3 throws.
\hfill \mbox{\textit{CAIE S1 2016 Q2 [5]}}