G Fowler, February 26, 1996.
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We are conducting a commando raid. There are 12 members of our group.
1. Their sentry has a probability of detection of 0.40.
(a) What is the probability that he detects 2 of our group?
(b) What is the probability that he detects 2 or fewer?
(c) What is the probability no one is detected?
2. Our group contained 4 demolition members. Suppose that 6 of our 12 members get through their lines.
(a) What is the probability that we have 2 demolition members?
(b) What is the probability none of our demolition members makes it?
(c) What is the probability of 2 or more making it?
We follow up with a larger force.
3. The sentry still has detection probability 0.40. What is the expected number that can penetrate their lines before the first one is detected?
4. Suppose the sentry ignores the first invader (thinks it's an illusion), but will call in when he sees the second one. What is the expected number that can get through before the sentry calls in? (Include the second person detected in your count.)
5. What is the probability that 8 invaders can get in before the sentry calls?
6. Land mines are laid along a road so that the average spacing is 10 feet (this means 0.1 mine/ft). What is the probability of no land mines in an 8 foot section?
7. Our new tank can withstand 2 (or fewer) hits. What is the probability it can travel 50 feet along the road?
8. Mortar fire comes at the rate of 3.1 rounds/minute.
(a) What is the probability of no rounds in a minute?
(b) What is the probability of no rounds in a half minute?
(c) Is it more likely that a randomly chosen minute will contain 2 rounds or 3 rounds? What are the probabilities?
9. While they are waiting for battle, they play cards.
(a) What is the probability that a 5 card hand contains exactly 3 spades?
(b) What is the probability of 0 spades?
(c) What is the probability of 2 or fewer spades?
Answers:
1. P(2) = 0.0638523
p(<=2) =0.0834433
p(0) = 0.0021768
2. P(2) = 0.4545455
p(0) = 0.030303
p(>=2) = 0.7272727
3. E(X)=2.5
4. E(X) = 2*2.5 =5
5. p(Exactly 8)=0.0522547
6. exp(-0.8) = 0.449329
7. p(<=2, rate=5) = 0.124652
8. p(0 in min) = 0.0450492
p(0 in 1/2 min) = 0.212248
p(2)=0.2164614 p(3)=0.2236768
9. p(3)=0.0815426
p(0) = 0.2215336
p(<=2) = 0.9072329