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IAL 2019 Oct Q3

A Level / Edexcel / S1

IAL 2019 Oct Paper · Question 3

题目

Problem

The distance that Jenny throws a javelin is normally distributed with mean 42 m and standard deviation 5 m.

(a) Find the probability that the distance Jenny throws the javelin is less than 40 m.

(3)

Jenny enters a javelin competition. To qualify for the final she has 3 attempts to throw the javelin a distance of more than 40 m. Once she has qualified she does not use any of her remaining attempts.

Given that Jenny qualified for the final and that throws of the javelin are independent,

(b) find the probability that she qualified for the final on her third throw with a distance greater than 45 m.

(5)

解答

(a)

解法一

思路

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先把 40 标准化，然后查左尾概率。

答题过程

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Let XX be the distance thrown.

X∼N(42,52).\begin{align*} X\sim N(42,5^2). \end{align*}

Then

P(X<40)=P(Z<40−425)=P(Z<−0.4).\begin{align*} P(X<40)=P\left(Z<\frac{40-42}{5}\right)=P(Z<-0.4). \end{align*}

Using symmetry,

P(Z<−0.4)=1−P(Z<0.4)=1−0.6554.\begin{align*} P(Z<-0.4)=1-P(Z<0.4)=1-0.6554. \end{align*}

Therefore

P(X<40)=0.3446.\begin{align*} P(X<40)=0.3446. \end{align*}

(b)

解法一

思路

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题目已经给定 Jenny qualified，所以分母是三次内至少一次超过 40 的概率。分子是前两次都不超过 40，第三次超过 45。

答题过程

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From part (a),

P(X≤40)=0.3446.\begin{align*} P(X\leq40)=0.3446. \end{align*}

So the probability that Jenny qualifies is

P(qualifies)=1−(0.3446)3=0.9590….\begin{align*} P(\text{qualifies})=1-(0.3446)^3=0.9590\ldots. \end{align*}

Also,

P(X>45)=P(Z>45−425)=P(Z>0.6).\begin{align*} P(X>45)=P\left(Z>\frac{45-42}{5}\right)=P(Z>0.6). \end{align*}

Thus

P(X>45)=1−0.7257=0.2743.\begin{align*} P(X>45)=1-0.7257=0.2743. \end{align*}

The probability that she qualifies on her third throw with a distance greater than 45 m is

(0.3446)2(0.2743).\begin{align*} (0.3446)^2(0.2743). \end{align*}

Therefore the required conditional probability is

(0.3446)2(0.2743)0.9590…=0.0339….\frac{(0.3446)^2(0.2743)}{0.9590\ldots} =0.0339\ldots.

So the probability is approximately

0.034.\begin{align*} 0.034. \end{align*}