题目

Problem

Charlie is training for three events: a 1500 m swim, a 40 km bike ride and a 10 km run.

From past experience his times, in minutes, for each of the three events independently have the following distributions.

S\sim N(41,5.2^2) \text{ represents the time for the swim}

B\sim N(81,4.2^2) \text{ represents the time for the bike ride}

R\sim N(57,6.6^2) \text{ represents the time for the run}

(a) Find the probability that Charlie’s total time for a randomly selected swim, bike ride and run exceeds 3 hours.

(b) Find the probability that the time for a randomly selected swim will be at least 20 minutes quicker than the time for a randomly selected run.

Given that P(S + B + R > t) = 0.95

A triathlon consists of a 1500 m swim, immediately followed by a 40 km bike ride, immediately followed by a 10 km run. Charlie uses the answer to part (a) to find the probability that, in 6 successive independent triathlons, his time will exceed 3 hours on at least one occasion.

(d) Find the answer Charlie should obtain.

Jane says that Charlie should not have used the answer to part (a) for the calculation in part (d).

(e) Explain whether or not Jane is correct.

(5)

(4)

(3)

(2)

(Total for Question 5 is 17 marks)

题目中文翻译

Charlie 正在为三个项目训练：1500 米游泳、40 公里自行车和 10 公里跑步。

根据过往经验，这三个项目所需时间（分钟）彼此独立，分别服从如下分布。

S\sim N(41,5.2^2) \text{ 表示游泳时间}

B\sim N(81,4.2^2) \text{ 表示骑车时间}

R\sim N(57,6.6^2) \text{ 表示跑步时间}

(a) 求 Charlie 随机选择一次游泳、一次骑车和一次跑步的总时间超过 3 小时的概率。

(b) 求随机一次游泳比随机一次跑步至少快 20 分钟的概率。

已知 $P(S+B+R>t)=0.95$ 。

三项全能赛包括 1500 米游泳、紧接着 40 公里骑车、紧接着 10 公里跑步。 Charlie 用 (a) 小题的答案来求在 6 次连续独立的三项全能赛中，他的总时间至少有一次超过 3 小时的概率。

(d) 求 Charlie 应得到的答案。

Jane 说，Charlie 在 (d) 小题的计算中不应该使用 (a) 小题的答案。

(e) 解释 Jane 是否正确。

IAL 2022 Jan S3 Q5

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