| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2016 |
| Session | Specimen |
| Marks | 6 |
| Topic | Binomial Distribution |
| Type | Verify conditions in context |
| Difficulty | Moderate -0.8 This question tests basic understanding of binomial distribution assumptions and a straightforward probability calculation. Part (i) requires recall of standard conditions (fixed n, constant p, independence), while part (ii) is a routine calculation of P(X ≤ 2) using B(20, 0.13). No problem-solving insight or complex manipulation is needed—both parts are textbook exercises testing foundational knowledge. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
**Question 4**
(i) Independence between children [B1]
Class is typical of population in respect of left-handedness [B1]
(ii) $13\%$ of $20 = 2.6$, so want $P(X \leqslant 2)$ [B1]
$(0.87)^{20} + 20(0.13)(0.87)^{19} + 190(0.13)^2(0.87)^{18}$
At least one probability in $B(20, 0.13)$ [M1]
$= 0.061714 + 0.18443 + 0.26181$ [A1]
$= 0.50795\ldots = 0.508$ to 3sf [A1]
**Total: 6 marks**4 A survey into left-handedness found that 13\% of the population of the world are left-handed.\\
(i) State the assumptions necessary for it to be appropriate to model the number of left-handed children in a class of 20 children using the binomial distribution $\mathrm { B } ( 20,0.13 )$.\\
(ii) Assuming that this binomial model is appropriate, calculate the probability that fewer than $13 \%$ of the 20 children are left-handed.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2016 Q4 [6]}}