| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Binomial with complementary events |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard binomial probability calculations and a basic hypothesis test. Parts (a)-(c) involve routine binomial probability computations with clear parameters, while part (d) is a textbook one-tailed binomial test requiring standard procedure (hypotheses, test statistic, critical region, conclusion). All techniques are standard S2 content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks |
|---|---|
| (a) Let \(X =\) no. of dice showing 6 \(\therefore X \sim B(6, \frac{1}{6})\); \(P(X = 0) = (\frac{5}{6})^6 = 0.3349\) (4sf) | M1 M1 A1 |
| (b) \(P(X > 1) = 1 - P(X \leq 1) = 1 - [(\frac{5}{6})^6 + 6(\frac{1}{6})(\frac{5}{6})^5] = 1 - 0.7368 = 0.2632\) (4sf) | M1 M1 A1 |
| (c) Let \(Y =\) no. of dice showing odd \(\therefore Y \sim B(6, \frac{1}{2})\); \(P(Y = 3) = 0.6563 - 0.3438 = 0.3125\) | M1 M1 A1 |
| (d) Let \(S =\) no. of times it shows a 6 \(\therefore S \sim B(8, \frac{1}{6})\); \(H_0: p = \frac{1}{6}\) \(H_1: p > \frac{1}{6}\); \(P(S \geq 3) = 1 - P(S \leq 2) = 1 - [(\frac{5}{6})^8 + 8(\frac{1}{6})(\frac{5}{6})^7 + \frac{8 \times 7}{2}(\frac{1}{6})^2(\frac{5}{6})^6] = 1 - 0.8652 = 0.1348\); more than 5% \(\therefore\) not significant, insufficient evidence of bias | B1 M1 M1 A1 A1 A1 |
(a) Let $X =$ no. of dice showing 6 $\therefore X \sim B(6, \frac{1}{6})$; $P(X = 0) = (\frac{5}{6})^6 = 0.3349$ (4sf) | M1 M1 A1 |
(b) $P(X > 1) = 1 - P(X \leq 1) = 1 - [(\frac{5}{6})^6 + 6(\frac{1}{6})(\frac{5}{6})^5] = 1 - 0.7368 = 0.2632$ (4sf) | M1 M1 A1 |
(c) Let $Y =$ no. of dice showing odd $\therefore Y \sim B(6, \frac{1}{2})$; $P(Y = 3) = 0.6563 - 0.3438 = 0.3125$ | M1 M1 A1 |
(d) Let $S =$ no. of times it shows a 6 $\therefore S \sim B(8, \frac{1}{6})$; $H_0: p = \frac{1}{6}$ $H_1: p > \frac{1}{6}$; $P(S \geq 3) = 1 - P(S \leq 2) = 1 - [(\frac{5}{6})^8 + 8(\frac{1}{6})(\frac{5}{6})^7 + \frac{8 \times 7}{2}(\frac{1}{6})^2(\frac{5}{6})^6] = 1 - 0.8652 = 0.1348$; more than 5% $\therefore$ not significant, insufficient evidence of bias | B1 M1 M1 A1 A1 A1 |
---Six standard dice with faces numbered 1 to 6 are thrown together.
Assuming that the dice are fair, find the probability that
\begin{enumerate}[label=(\alph*)]
\item none of the dice show a score of 6, [3 marks]
\item more than one of the dice shows a score of 6, [4 marks]
\item there are equal numbers of odd and even scores showing on the dice. [3 marks]
\end{enumerate}
One of the dice is suspected of being biased such that it shows a score of 6 more often than the other numbers. This die is thrown eight times and gives a score of 6 three times.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Stating your hypotheses clearly, test at the 5% level of significance whether or not this die is biased towards scoring a 6. [7 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q5 [17]}}