| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Topic | Binomial Distribution |
| Type | E(X) and Var(X) with probability calculations |
| Difficulty | Moderate -0.8 This is a straightforward binomial distribution question requiring only standard calculations: finding the mean (np), computing P(X=8) using the binomial formula, and finding P(X≥8) = 1-P(X≤7). All three parts are routine textbook exercises with no problem-solving or conceptual challenges, making it easier than average but not trivial since it requires correct application of binomial probability formulas. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.02d Binomial: mean np and variance np(1-p) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(E(X) = 20 \times 0.4 = 8\) | B1 [1] | |
| (ii) State or imply \(\text{Bin}(20, 0.4)\) | B1 | May be awarded elsewhere if not here. |
| \(P(X = 8) = 0.5956 - 0.4159 = 0.1797\) | M1, A1 [3] | Use of tables for \(P(X \leq 8) - P(X \leq 7)\) or formula for \(P(X = 8)\). c.a.o. |
| (iii) \(P(X \geq 8) = 1 - 0.4159 = 0.5841\) | M1, A1 [2] | Attempt 1 – \(P(X \leq 7)\). c.a.o. |
**(i)** $E(X) = 20 \times 0.4 = 8$ | B1 [1] |
**(ii)** State or imply $\text{Bin}(20, 0.4)$ | B1 | May be awarded elsewhere if not here.
$P(X = 8) = 0.5956 - 0.4159 = 0.1797$ | M1, A1 [3] | Use of tables for $P(X \leq 8) - P(X \leq 7)$ or formula for $P(X = 8)$. c.a.o.
**(iii)** $P(X \geq 8) = 1 - 0.4159 = 0.5841$ | M1, A1 [2] | Attempt 1 – $P(X \leq 7)$. c.a.o.In a certain country 40\% of the population have brown eyes. A random sample of 20 people is chosen from that population.
\begin{enumerate}[label=(\roman*)]
\item Find the expected number of people in the sample who have brown eyes. [1]
\item Find the probability that there are exactly 8 people with brown eyes in the sample. [3]
\item Find the probability that there are at least 8 people with brown eyes in the sample. [2]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2014 Q4 [6]}}