| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2015 |
| Session | June |
| Marks | 12 |
| Topic | Binomial Distribution |
| Type | Two-stage binomial problems |
| Difficulty | Standard +0.3 This is a straightforward two-stage binomial problem requiring identification of B(3, 0.7), calculation of standard binomial probabilities, recognition that 'at most 1 seedling' creates a modified distribution, and a final compound probability calculation. All steps are routine applications of binomial distribution with clear structure and no novel insight required. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\) | 0 | 1 |
| \(P(X = x)\) | 0.027 | 0.973 |
**Question 5**
**(i)** $(X \sim)\ \text{Bin}(3,\ 0.7)$
— B2: All 3 elements present and correct. Allow B1 for only 1 error/omission. **[2]**
**(ii)(a)** $P(X = 2) = 3 \times 0.7^2 \times 0.3$
$= 0.441$
— M1: $^3C_2 \times \ldots$ M1: $\ldots p^2 \times q$. A1: c.a.o. **[3]**
**(ii)(b)** $P(X \geqslant 1) = 1 - 0.3^3$
$= 0.973$
— M1: Or by summing $P(1) \ldots P(3)$. A1: c.a.o. **[2]**
**(iii)**
| $x$ | 0 | 1 |
|---|---|---|
| $P(X = x)$ | 0.027 | 0.973 |
— B1: Values in top row. B1: $P(X=1) = \textit{their } \textbf{(ii)(b)}$. B1: $P(X=0) = 1 - \textit{their } \textbf{(ii)(b)}$. $\text{Bin}(1,\ 0.973)$ earns full marks. **[3]**
**(iv)** $P(\text{All contain a seedling}) = 0.973^6$
$= 0.84854\ldots \approx 0.849$
— M1: Ft *their* $P(X=1)$. A1: c.a.o. **[2]**5 A garden centre grows a particular variety of plant for sale. They sow 3 seeds in each pot and there are 6 pots in a tray. The probability that a seed germinates is 0.7 , independently of any other seeds.\\
(i) State the probability distribution of the number of seeds in a pot that germinate.\\
(ii) Find the probability that, in a randomly chosen pot,
\begin{enumerate}[label=(\alph*)]
\item exactly 2 seeds germinate,
\item at least 1 seed germinates.
After the seeds have germinated and become seedlings, some are removed (and discarded) so that there remains at most 1 seedling per pot.\\
(iii) Write out the probability distribution of the number of seedlings per pot that remain.\\
(iv) Find the probability that there is a seedling in every one of the 6 pots in a randomly chosen tray.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2015 Q5 [12]}}