10/21/96 - 1001, 2001

You may use your calculators. Specify which program you used and with what parameters. For example, BCDF(p=4/9,N=17,X=3). Also, circle your final answer and give 4 decimal places. The setting is a RAF base in England during World War II.

1. Some replacement pilots have finally arrived. Only about 35% of new pilots are night-qualified and we need 5 new night-qualified pilots.
   1. If we receive 20 new pilots, what is the chance of getting the 5 night-qualified pilots we need?
   2. If we want to be 99% sure to get 5 night-qualified pilots, how many replacements should we ask for?
2. One of our targets is such that we will need to make 3 separate strikes. The weather is bad and there is only a 40% of any strike getting through.
   1. If we only have time to schedule 7 strikes, what is the chance that enough of these will get through?
   2. How many strikes must we schedule in order to be 80% sure of enough getting through?
3. Our stockpile of bombs contains 15 live rounds and 7 dummies (which are not otherwise indistinguishable). We need 4 live rounds and so tell our crew to fetch 6 from the stockpile.
   1. What is the chance that we get the live rounds we need?
   2. To be 85% sure of getting 4 live rounds, how many should we have fetched?
4. A bombing mission will last 8 hours. A fighter can go (on average) 10 hours without a failure.
   1. What is the probability that a fighter can last for the entire bombing mission?
   2. Use your answer in part (a) to determine the probability that we will finish with at least 5 fighters if we start with 8?
   3. We want to have at least a 90% chance of finishing with 5 fighters when we start with 8. What is the minimum value (to 2 decimal places) for the probability of each fighter finishing a mission?
5. Some of the incoming bombers have been modified to have rear gunners. We estimate that about 40% are so modified. Our fighters attack 20 bombers. What is the chance that more than 10 have rear gunners?
6. This is a joint operation, with 60% British and 40% American. If I select a bomber crew of 8 at random from my force, what is the chance of having at least 3 British and at least 2 Americans?
7. The radar sets were not very reliable back then. The average time between failures was only 4 hours. Therefore, planes carried 3 sets (1 to use and 2 for backups when the others failed).
   1. What is the chance we can finish an 8 hour mission with our radar working?
   2. How many total sets should we carry to be 90% sure that we finish the mission with working radar?