A dice manufacturer claims that for a novelty die he produces the probability of scoring thenumbers 1 to 5 are all equal, and the probability of a 6 is two times the probability ofscoring any of the other numbers.To test the manufacture’s claim one of the novelty dice is rolled 350 times and the numbersscored on the die are shown in the table below.A χ2 goodness of fit test is to be used with a 5% significance level.

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

Let the probability of scoring$1,\dots ,5$be$p,$

$5p+2p=1\Rightarrow p=\frac{1}{7}$ (M1)(A1)

Probability of$6=\frac{2}{7}$ A1

[3 marks]

Let the number of sixes be$X$

$X~\text{B}\left(5,\frac{2}{7}\right)$ (M1)

$\text{P}\left(X>2\right)=\text{P}\left(X\ge 3\right)$or$\text{P}\left(X>2\right)=1-\text{P}\left(X\le 2\right)$ (M1)

$=0.145\left(0.144701\dots \right)$ (M1)A1

[4 marks]

Expected frequency is$350\times p$or$350\times 2p$ (M1)

A1

[2 marks]