| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Verify conditions in context |
| Difficulty | Moderate -0.8 This question tests basic understanding of binomial distribution conditions and straightforward probability calculations. Part (a) is pure recall, part (b) requires recognizing that sampling without replacement violates independence, and parts (c)-(e) involve standard binomial probability calculations and expectation using formulas directly. The conceptual insight needed is minimal and the calculations are routine for S2 level. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(n\) repeated experiments, each with the same two possible outcomes with constant probabilities of success or fail | B2 | |
| (b) If cards not replaced, probabilities change after each draw | B2 | |
| (c) Replace card each time, to get \(B(10, 0.25)\) | B1, M1, A1 | |
| (d) \(P(X = 0) = 0.7759 - 0 = 0.224\) | M1, A1 | |
| (e) \(500 \times 0.25 = 125\) | M1, A1 | 11 marks total |
(a) $n$ repeated experiments, each with the same two possible outcomes with constant probabilities of success or fail | B2 |
(b) If cards not replaced, probabilities change after each draw | B2 |
(c) Replace card each time, to get $B(10, 0.25)$ | B1, M1, A1 |
(d) $P(X = 0) = 0.7759 - 0 = 0.224$ | M1, A1 |
(e) $500 \times 0.25 = 125$ | M1, A1 | 11 marks total |\begin{enumerate}[label=(\alph*)]
\item Briefly describe the main features of a binomial distribution. [2 marks]
\end{enumerate}
I conduct an experiment by randomly selecting 10 cards, without replacement, from a normal pack of 52.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Explain why the distribution of $X$, the number of hearts obtained, is not $\text{B}(10, \frac{1}{4})$. [2 marks]
\end{enumerate}
After making the appropriate adjustment to the experiment, which should be stated, so that the distribution is $\text{B}(10, \frac{1}{4})$, find
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item the probability of getting no hearts, [3 marks]
\item the probability of getting 4 or more hearts. [2 marks]
\item If the modified experiment is repeated 50 times, find the total number of hearts that you would you expect to have drawn. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q3 [11]}}