Question

The probability that a missile hits a target successfully is 0.75. In order to destroy the target

completely, at least three successful hits are required. Then the minimum number of missiles that

have to be fired so that the probability of completely destroying the target is NOT less than 0.95, is​

Answer 1

Given : The probability that a missile hits a target successfully is 0.75. In order to destroy the target  completely, at least three successful hits are required.

To Find :  minimum number of missiles that  have to be fired so that the probability of completely destroying the target is NOT less than 0.95

Solution:

Let say number of missiles fired = n

Probability that a missile hits a target successfully  p = 0.75

missile does not hit target q  = 1 - p = 1 -0.75 = 0.25

P(x) = ⁿCₓpˣqⁿ⁻ˣ

three successful hits

1 -P(0) - P(1) - P(2)  > 0.95

=> P(0) + P(1) + P(2) < 0.05

ⁿC₀(0.75)⁰(0.25)ⁿ + ⁿC₁(0.75)¹(0.25)ⁿ⁻¹ +  ⁿC₂(0.75)²(0.25)ⁿ⁻² < 0.05

=ⁿC₀(0.75)⁰(0.25)ⁿ  (1  + 3n +  9n(n-1)/2) < 0.05

= (0.25)ⁿ (9n² - 3n  +  2) < 0.1  

n = 5

= 0.207  > 0.1

n = 6

= 0.075 <    0.1

Minimum 6 number of missiles that  have to be fired so that the probability of completely destroying the target is NOT less than 0.95

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Answer 2

The probability that a missile hits a target successfully is 0.75. In order to destroy the target

completely, at least three successful hits are required. Then the minimum number of missiles that

have to be fired so that the probability of completely destroying the target is NOT less than 0.95, is