题目

Problem

In a certain lottery, on average $1$ in every $10\,000$ tickets is a prize-winning ticket. An agent sells $6000$ tickets.

(a) Use a suitable approximating distribution to find the probability that at least $3$ of the tickets sold by the agent are prize-winning tickets.

(3)

(b) Justify the use of your approximating distribution in this context.

(1)

题目中文翻译

在某种彩票中，平均每 $10\,000$ 张彩票有 $1$ 张中奖。一名代理商售出 $6000$ 张彩票。

(a) 用适当的近似分布求该代理商售出的彩票中至少有 $3$ 张中奖的概率。

(3)

(b) 结合题意说明采用该近似分布的理由。

(1)

原卷通用作答要求

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

解答

(a)

解法一

思路

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大量彩票、低中奖率，采用参数6000乘1/10000的泊松近似，至少3张取0至2张的补集。

答题过程

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Let $W$ be the number of prize-winning tickets sold. Using the Poisson approximation,

W\approx\mathrm{Po}\bigg(6000\frac1{10000}\bigg)=\mathrm{Po}(0.6).

Therefore

\begin{align*} \mathrm{P}(W\geq3)\approx&\,1-e^{-0.6}\bigg(1+0.6+\frac{0.6^2}{2!}\bigg)\\[4mm] =&\,0.0231153\ldots\\[4mm] =&\,\boxed{0.0231}\quad\text{(3 s.f.).} \end{align*}

(b)

解法一

思路

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结合彩票情境给出具体的大n、小p参数。

答题过程

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The agent sells many tickets, $n=6000>50$ , and the probability that any ticket wins is small, $p=\frac1{10000}<0.1$ , so a Poisson approximation is suitable.

CIE 9709 2024 March Paper 62 Q3

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程