Question

Suppose that a satellite defense system is established with five satellites acting independently. Individually, each satellite has a 0.85 probability of detecting an incoming ballistic missile. What is the probability that at least one of the five satellites detect an incoming ballistic missile? Round to four decimal places.

Answer 1

Solution :

Let X be a random variable which represents that out of 5 satellites, number of satellites that detect an incoming ballistic missile.

Let's consider "detection of incoming ballistic missile by a satellite" as success. So, we have only two mutually exclusive outcomes (success and failure) for each of the trials.

Probability of success (p) = 0.85

Number of trials (n) = 5

Since, the probability of success remains constant in each of the trial, outcomes are independent, number of trials are finite, therefore we can consider that X follows binomial distribution with parameters n = 5 and p = 0.85.

We have to find P(X = at least 1).

According to binomial probability law, probability of occurrence of exactly x successes in n trials is given by,

Where, p is probability of success.

We have, n = 5 and p = 0.85

P(X = at least 1) = P(X ≥ 1)

P(X = at least 1) = 1- P(X < 1)

P(X = at least 1) = 1 - P(X = 0)

Using binomial probability law we get,

Hence, the probability that at least one of the five satellites detect an incoming ballistic missile is 0.9999.

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