Comparing two journey times

2 Each day Samuel travels from \(A\) to \(B\) and from \(B\) to \(C\). He then returns directly from \(C\) to \(A\). The times, in minutes, for these three journeys have the independent distributions \(\mathrm { N } \left( 20,2 ^ { 2 } \right) , \mathrm { N } \left( 18,1.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 30,1.8 ^ { 2 } \right)\), respectively. Find the probability that, on a randomly chosen day, the total time for his two journeys from \(A\) to \(B\) and \(B\) to \(C\) is less than the time for his return journey from \(C\) to \(A\). [5]

6 The times, in minutes, taken to complete the two parts of a task are normally distributed with means 4.5 and 2.3 respectively and standard deviations 1.1 and 0.7 respectively.

1. Find the probability that the total time taken for the task is less than 8.5 minutes.
2. Find the probability that the time taken for the first part of the task is more than twice the time taken for the second part.

2. Philip and James are racing car drivers. Philip's lap times, in seconds, are normally distributed with mean 90 and variance 9. James' lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap.

1. Find the probability that James' time for the qualifying lap is less than Philip's. The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent,
2. find the probability that Philip beats James in the race by more than 2 minutes.

6 An aircraft, based at airport A, flies regularly to and from airport B.
The aircraft's flying time, \(X\) minutes, from A to B has a mean of 128 and a variance of 50 .
The aircraft's flying time, \(Y\) minutes, on the return flight from B to A is such that $$\mathrm { E } ( Y ) = 112 , \quad \operatorname { Var } ( Y ) = 50 \quad \text { and } \quad \rho _ { X Y } = - 0.4$$

1. Given that \(F = X + Y\) :
   1. find the mean of \(F\);
   2. show that the variance of \(F\) is 60 .
2. At airport B , the stopover time, \(S\) minutes, is independent of \(F\) and has a mean of 75 and a variance of 36 . Find values for the mean and the variance of:
   1. \(T = F + S\);
   2. \(M = F - 3 S\).
3. Hence, assuming that \(T\) and \(M\) are normally distributed, determine the probability that, on a particular round trip of the aircraft from A to B and back to A :
   1. the time from leaving A to returning to A exceeds 300 minutes;
   2. the stopover time is greater than one third of the total flying time.

7 Two flatmates work at the same location. One of them takes the bus to work and the other one cycles. Journey times, measured in minutes, are distributed as follows.

• By bus: Normally distributed with mean 23 and standard deviation 6
• By bicycle: Normally distributed with mean 21 and standard deviation 2

You should assume that all journey times are independent.

1. One morning the two flatmates set out at the same time. Find the probability that the person who takes the bus arrives before the cyclist.
2. Find the probability that the total time taken for 5 bus journeys is less than 2 hours.
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