Problem A a) There are 7 shots in a team. Each of them hits some target with pro

Question

Problem A

a) There are 7 shots in a team. Each of them hits some target with probability 0.3. They shoot simultaneously and independently of each other. Find the probability that at least one of them will hit the target.

b) Find the probability that the target will be hit at least once if the team includes8 shots.c) Suppose the probability of hitting the target by a team (at least once) has to be not less than 0.97. Each shot hits the target with probability 0.3. How big has to be the team to fulfill this condition?Hint to Part

c). You have to solve an inequality of the following kind:pnPforn(here both and P are probabilities, so ln p<0). Keep in mind that the sense of the inequality changes when you divide both parts of it by a negative number.

Explanation / Answer

The chance of hitting the target at least once is the inverse probability of never hitting it.

The probability to not hit on a single shot is 1-0.3 = 0.7

The probability to never hit on n shots is 0.7^n

The probability to hit at least once in n shots is then:
P(at least one) = 1-0.7^n

a) In 7 shots: P(at least one in seven) = 1-0.7^7 = 91.7%

b) In 8 shots: P(at least one in eight) = 1-0.7^8 = 94.2%

c) You are looking for n verifying: 1-0.7^n 0.97

that means 0.7^n 0.03

n*log(0.7) log(0.03)

n log(0.03)/log(0.7) (invert the inequality because you are dividing both sides with log(0.7), a negative number)

log(0.03)/log(0.7) = 9.83

You need at least 10 tries.

Verification: P(at least one in 9) = 1-0.7^9 = 95.9%
P(at least one in 10) = 1-0.7^10 = 97.17%

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