| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2003 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | State general binomial conditions |
| Difficulty | Moderate -0.3 Part (a) is pure recall of binomial conditions (4 marks of bookwork). Parts (b)(i-ii) require calculating p from the bias condition (straightforward: p=2/7), then applying standard formulas - geometric distribution for first success and binomial probability. The bias setup is simple arithmetic, and both probability calculations are direct formula applications with no conceptual challenges beyond S2 syllabus expectations. |
| Spec | 2.04b Binomial distribution: as model B(n,p)5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| Fixed number of independent trials | B1 B1 | |
| 2 outcomes | B1 | |
| Probability of success constant | B1 | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X = 5) = \frac{2}{7}\); \(P(X \neq 5) = \frac{5}{7}\) | may be implied | B1; B1 ft |
| \(P(5 \text{ on sixth throw}) = \left(\frac{5}{7}\right)^5 \times \left(\frac{2}{7}\right)\) | \(p^n(1-p)\) | M1 A1 ft |
| \(= 0.0531\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(\text{exactly 3 fives in first eight throws}) = \binom{8}{3}\left(\frac{2}{7}\right)^3\left(\frac{5}{7}\right)^3\) | use of "\(C_r\) needed" | M1 A1 ft |
| \(= 0.243\) | A1 |
## (a)
Fixed number of independent trials | B1 B1 |
2 outcomes | B1 |
Probability of success constant | B1 | (4)
## (b)
$P(X = 5) = \frac{2}{7}$; $P(X \neq 5) = \frac{5}{7}$ | may be implied | B1; B1 ft |
$P(5 \text{ on sixth throw}) = \left(\frac{5}{7}\right)^5 \times \left(\frac{2}{7}\right)$ | $p^n(1-p)$ | M1 A1 ft |
$= 0.0531$ | | A1 | (5)
## (c)
$P(\text{exactly 3 fives in first eight throws}) = \binom{8}{3}\left(\frac{2}{7}\right)^3\left(\frac{5}{7}\right)^3$ | use of "$C_r$ needed" | M1 A1 ft |
$= 0.243$ | | A1 | (3)
---\begin{enumerate}[label=(\alph*)]
\item Write down the conditions under which the binomial distribution may be a suitable model to use in statistical work. [4]
\end{enumerate}
A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly.
Find the probability that
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item \begin{enumerate}[label=(\roman*)]
\item the first 5 will occur on the sixth throw,
\item in the first eight throws there will be exactly three 5s.
\end{enumerate} [8]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2003 Q4 [12]}}