2 A manufacturer is testing how long coloured LED lights will last before the battery runs out, using two different battery types. The times in hours before the battery runs out are modelled by independent Normal distributions with means and standard deviations as shown in the table.
| \cline { 2 - 3 }
\multicolumn{1}{c|}{} | Time |
| Type | Mean | |
| A | 23 | 2.8 |
| B | 35 | 3.6 |
- In a particular test, a battery of type A is used and the time taken for it to run out is recorded. This process is repeated until a total of 5 randomly selected batteries have been used.
Determine the probability that the total time the 5 batteries last is at least 120 hours.
- In a similar test, 3 randomly selected batteries of type A are used, one after the other. Then 2 randomly selected batteries of type B are used, one after the other.
Determine the probability that the 3 type A batteries last longer in total than the 2 type B batteries.
- Explain why it is necessary that the Normal distributions are independent in order to be able to find the probability in part (b).