| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | One unknown from sum constraint only |
| Difficulty | Easy -1.3 This is a straightforward discrete probability distribution question requiring only basic probability axioms. Part (i) uses the sum-to-one constraint to find k (simple algebra), part (ii) identifies the mode by inspection, and part (iii) requires calculating the mean and summing probabilities—all routine procedures with no problem-solving insight needed. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables |
| \(x\) | 0 | 1 | 2 | 3 | 4 | \(\geqslant 5\) |
| \(\mathrm { P } ( X = x )\) | 0.24 | 0.35 | \(2 k\) | \(k\) | 0.05 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(0.24 + 0.35 + 2k + k + 0.05 = 1\); \(k = 0.12\) | M1 A1 | Summing probs = 1; Correct answer |
| 2 marks | ||
| (ii) Model number is 1 | B1 | 1 mark |
| (iii) \(\text{mean} = 1 \times 0.35 + 2 \times 0.24 + 3 \times 0.12 + 4 \times 0.05\); \(P(X > 1.39) = P(2, 3, 4) = 0.41\) | B1 M1 B1 | 1.39 seen; Finding \(P(X >\) their mean); Correct ans following mean or mode only |
| 3 marks |
**(i)** $0.24 + 0.35 + 2k + k + 0.05 = 1$; $k = 0.12$ | M1 A1 | Summing probs = 1; Correct answer |
| | | 2 marks |
**(ii)** Model number is 1 | B1 | 1 mark |
**(iii)** $\text{mean} = 1 \times 0.35 + 2 \times 0.24 + 3 \times 0.12 + 4 \times 0.05$; $P(X > 1.39) = P(2, 3, 4) = 0.41$ | B1 M1 B1 | 1.39 seen; Finding $P(X >$ their mean); Correct ans following mean or mode only |
| | | 3 marks |
---2 The number of phone calls, $X$, received per day by Sarah has the following probability distribution.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
$x$ & 0 & 1 & 2 & 3 & 4 & $\geqslant 5$ \\
\hline
$\mathrm { P } ( X = x )$ & 0.24 & 0.35 & $2 k$ & $k$ & 0.05 & 0 \\
\hline
\end{tabular}
\end{center}
(i) Find the value of $k$.\\
(ii) Find the mode of $X$.\\
(iii) Find the probability that the number of phone calls received by Sarah on any particular day is more than the mean number of phone calls received per day.
\hfill \mbox{\textit{CAIE S1 2014 Q2 [6]}}