| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Direct probability from given distribution |
| Difficulty | Standard +0.3 This is a straightforward discrete probability distribution question requiring basic probability calculations, variance formula application, and solving a simple equation. While it has three parts, each involves standard S1 techniques with no conceptual challenges—slightly easier than average due to the mechanical nature of the calculations. |
| Spec | 2.04a Discrete probability distributions5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X < 2) = 27/70 + 1/10 = 34/70 (17/35) (0.486)\) | B1 | 9/70 Seen |
| B1 [2] | FT their probs if < 1 | |
| (ii) \(E(X) = 108/70 (54/35) (1.543)\) | M1 | Valid attempt at \(E(X)\) (needn't be accurate) |
| \(\text{Var}(X) = ((-2)^2 + \ldots + 5^2) \times 9/70 - (54/35)^2 = 5.33\) | M1 | Using a variance formula correctly with mean\(^2\) subtracted numerically, no extra division |
| A1 [3] | Correct final answer | |
| (iii) \(a = 1\) | B1 [1] |
**(i)** $P(\text{any other number}) = 9/70$
$P(X < 2) = 27/70 + 1/10 = 34/70 (17/35) (0.486)$ | B1 | 9/70 Seen
| B1 [2] | FT their probs if < 1
**(ii)** $E(X) = 108/70 (54/35) (1.543)$ | M1 | Valid attempt at $E(X)$ (needn't be accurate)
$\text{Var}(X) = ((-2)^2 + \ldots + 5^2) \times 9/70 - (54/35)^2 = 5.33$ | M1 | Using a variance formula correctly with mean$^2$ subtracted numerically, no extra division
| A1 [3] | Correct final answer
**(iii)** $a = 1$ | B1 [1] |3 The possible values of the random variable $X$ are the 8 integers in the set $\{ - 2 , - 1,0,1,2,3,4,5 \}$. The probability of $X$ being 0 is $\frac { 1 } { 10 }$. The probabilities for all the other values of $X$ are equal. Calculate\\
(i) $\mathrm { P } ( X < 2 )$,\\
(ii) the variance of $X$,\\
(iii) the value of $a$ for which $\mathrm { P } ( - a \leqslant X \leqslant 2 a ) = \frac { 17 } { 35 }$.
\hfill \mbox{\textit{CAIE S1 2011 Q3 [6]}}