6. Four swimmers, \(A , B , C\) and \(D\), are to be used in a \(4 \times 100\) metres freestyle relay. The time for each swimmer to complete a leg follows a normal distribution. The mean and standard deviation, in seconds, of the time for each swimmer to complete a leg and the order in which they are to swim are shown in the table below.
| mean | standard deviation |
| \(1 ^ { \text {st } }\) leg \(- A\) | 63.1 | 1.2 |
| \(2 ^ { \text {nd } }\) leg \(- B\) | 65.7 | 1.5 |
| \(3 ^ { \text {rd } } \operatorname { leg } - C\) | 65.4 | 1.8 |
| \(4 ^ { \text {th } }\) leg - \(D\) | 62.5 | 0.9 |
- Find the probability that the total time for first two legs is less than the total time for the last two.
(6 marks)
The total time for another team to complete this relay is normally distributed with a mean of 259.0 seconds and a standard deviation of 3.4 seconds. The two teams are to compete over four races. - Find the probability that the first team wins all four races, assuming that the team's performances are not affected by previous results.
(8 marks)