| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Sum or product of two independent values |
| Difficulty | Moderate -0.3 This is a straightforward S1 question testing basic probability distribution properties (probabilities sum to 1), independence of events, and expected value linearity. Parts (a) and (d) are routine recall, while parts (b) and (c) require systematic enumeration of cases but no novel insight. Slightly easier than average due to small sample spaces and standard techniques. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(x\) | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | \(0 \cdot 2\) | \(0 \cdot 4\) | \(p\) |
| \(Y\) | 1 | 2 | 3 | 4 |
| \(\mathrm { P } ( Y = y )\) | 0.2 | 0.5 | \(q\) | \(q\) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(p = 0.4\), \(2q = 0.3\), \(q = 0.15\) | B1 B1 | |
| (b) Using sample space or otherwise | M1 | |
| (i) \(P(\text{sum} = 5) = 0.03 + 0.06 + 0.2 = 0.29\) | M1 A1 | |
| (ii) \(P(\text{sum} < 4) = 0.04 + 0.1 + 0.08 = 0.22\) | M1 A1 | |
| (c) Assumed independence. One is not likely to affect the other | B1 B1 | |
| (d) \(2(0.04) + 3(0.18) + 4(0.31) + 5(0.29) + 6(0.12) + 7(0.06) = 4.45\) | M1 M1 A1 A1 | 13 marks |
(a) $p = 0.4$, $2q = 0.3$, $q = 0.15$ | B1 B1 |
(b) Using sample space or otherwise | M1 |
(i) $P(\text{sum} = 5) = 0.03 + 0.06 + 0.2 = 0.29$ | M1 A1 |
(ii) $P(\text{sum} < 4) = 0.04 + 0.1 + 0.08 = 0.22$ | M1 A1 |
(c) Assumed independence. One is not likely to affect the other | B1 B1 |
(d) $2(0.04) + 3(0.18) + 4(0.31) + 5(0.29) + 6(0.12) + 7(0.06) = 4.45$ | M1 M1 A1 A1 | 13 marks\begin{enumerate}
\item Two spinners are in the form of an equilateral triangle, whose three regions are labelled 1,2 and 3, and a square, whose four regions are labelled $1,2,3$ and 4 . Both spinners are biased and the probability distributions for the scores $X$ and $Y$ obtained when they are spun are respectively:
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | | c | c | c | }
\hline
$x$ & 1 & 2 & 3 \\
\hline
$\mathrm { P } ( X = x )$ & $0 \cdot 2$ & $0 \cdot 4$ & $p$ \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | | l | l | l | l | }
\hline
$Y$ & 1 & 2 & 3 & 4 \\
\hline
$\mathrm { P } ( Y = y )$ & 0.2 & 0.5 & $q$ & $q$ \\
\hline
\end{tabular}
\end{center}
(a) Find the values of $p$ and $q$.\\
(b) Find the probability that, when the two spinners are spun together, the sum of the two scores is (i) 5, (ii) less than 4 .\\
(c) State an assumption that you have made in answering part (b) and explain why it is likely to be justifiable.\\
(d) Calculate $\mathrm { E } ( X + Y )$.\\
\hfill \mbox{\textit{Edexcel S1 Q5 [13]}}