2022 ACJC P2 Q9

In this question you should state clearly all the distributions that you use, together with the values of the appropriate parameters. You should also assume that \(T\) and \(X\) follow independent normal distributions.

Each KTX train takes \(T\) minutes to travel from Seoul Train Station to Pohang Train Station. It is known that \(T\) follows the distribution \(\mathrm{N}\left( 144,25 \right)\).

1. The probability that a randomly selected KTX train takes more than \(k\) hours to reach Pohang is \(0.5\). Without the use of a calculator, explain why \(k=2.4\).[1]

Before 7pm daily, each express bus takes \(X\) minutes to travel from Seoul Express Bus Terminal to Pohang Express Bus Terminal. It is known that \(X\) follows the distribution \(\mathrm{N}\left( 236,81 \right)\). After 7pm daily, the travel time taken by each express bus will be reduced by \(10\%\) as all the express buses will not make a stop at Daegu.

2. Find the probability that after 7pm, to travel from Seoul to Pohang, the travel time of a randomly selected KTX train is at most an hour faster than the travel time of a randomly selected express bus.[4]

Kim needs to travel from Seoul to Pohang on Saturday mornings to visit his parents. He prefers to take a KTX train if the tickets are available. On average, \(70\%\) of his journeys are by train.

3. On a particular Saturday morning, there was a train delay of \(1.6\) hours due to a train fault. Given that Kim took more than \(4\) hours to reach Pohang on that morning, find the probability that Kim travels to Pohang by train.[3]

The cost of taking a KTX train and taking an express bus from Seoul to Pohang are \(\$54\) and \(\$24\) respectively.

4. Find Kim’s expected cost of travelling (one-way) from Seoul to Pohang.[1]