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)

题目中文翻译

Jenny 投掷标枪的距离服从平均数为 42 m、标准差为 5 m 的正态分布。

(a) 求 Jenny 投掷标枪的距离小于 40 m 的概率。

Jenny 参加标枪比赛。她有 3 次机会把标枪投出超过 40 m；一旦取得决赛资格，她便不会使用余下的机会。

已知 Jenny 已取得决赛资格，且各次投掷互相独立。

(b) 求她在第三次投掷时以超过 45 m 的成绩取得决赛资格的概率。

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 $X$ be the distance thrown.

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

Then

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

Using symmetry,

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

Therefore

\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),

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

So the probability that Jenny qualifies is

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

Also,

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

Thus

\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

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

Therefore the required conditional probability is

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

So the probability is approximately

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

S1 2019 Oct 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