A missile protection system consists of n radar sets...

A missile protection system consists of $n$ radar sets operating independently, each with a probability of .9 of detecting a missile entering a zone that is covered by all of the units. a. If $n=5$ and a missile enters the zone, what is the probability that exactly four sets detect the missile? At least one set? b. How large must $n$ be if we require that the probability of detecting a missile that enters the zone be $.999 ?$

A missile protection system consists of $n$ radar sets operating independently, each with a probability of .9 of detecting a missile entering a zone that is covered by all of the units.
a. If $n=5$ and a missile enters the zone, what is the probability that exactly four sets detect the missile? At least one set?
b. How large must $n$ be if we require that the probability of detecting a missile that enters the zone be $.999 ?$

Key Concepts

Independent Events

This concept states that the outcome of one event does not affect the outcome of another. When dealing with multiple sensors or trials, knowing that each operates independently allows us to multiply individual probabilities to find the probability of combined outcomes.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials where each trial has the same probability of success. It provides the formula for computing the probability of obtaining exactly a given number of successes out of n trials.

Complement Rule

This rule is used to compute the probability of an event by subtracting the probability of its complement from one. For instance, to find the probability of at least one successful detection, one can subtract the probability of no successful detections from one.

Threshold Analysis Using Logarithms

When a required overall probability (or threshold) is specified, one can set up an inequality based on the cumulative probability (often using the complement rule) and solve for the minimum number of trials or units needed. This process typically involves taking logarithms to solve for the variable of interest.

Transcript

00:01 In question 48, we're looking at a problem in which we have radars in different sets trying to detect missiles.

00:07 It tells us that they all work independently of each other and they all work at 90 % efficiency.

00:17 We know that there's going to be five radars in a set trying to detect a missile.

00:22 In question a wants us to figure out what is the probability that exactly four of the five sets of radars detect a missile.

00:32 Now, this is a binomial setting, so we can use our binomial formula here, where we can just say if we have five possibilities, we need four successes.

00:42 So we're going to say 0 .9 raised to the fourth power times the failure, 0 .1 raised to the first power.

00:48 Or we could do this in our ti -84 calculator using the binomial pdf command in which we type in n to be 5.

00:57 Probability is 0 .9, and our x value is 4.

01:00 Either one of these will give you the same answer, which is 0 .3 -280.

01:05 The next part of a asks us to find the probability that x is at least one.

01:10 So there's at least one radar that detects the missile.

01:13 So we could find the probability of 1, 2, 3, 4, 5, and add them altogether.

01:18 Or we can use the complement rule that says 1 minus the probability of none is the same thing as the probability of at least 1.

01:27 So if we can figure out the probability of none, which is zero radars detect the missile, we can subtract that from one and figure out our answer...

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