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Combinatorics Problem

An over temperature warning system has five temperature sensors. To prevent false alarms, the system is designed so that three or more of the sensors must sense over temperature before a warning signal is given. Assume that the probability of any single sensor failing is equal likely and that three sensors are in a failed state. However, as the inspector, you are not aware of the failed sensors.

If three sensors are selected at random for inspection, what is the probability that the system will be found in a failed state? Please explain equation used and steps to solve the problem.

Solution Preview

First, let us reword the problem more simply:

There are five sensors. Two are working. Three have failed.
The inspector chooses three sensors at random.
Find: probability that all three have failed.

There are several ways to approach the problem. I will present two alternatives.

Solution 1:
In order to get three failed sensors, the inspector has to choose a failed sensor three times in a row.
The ...

Solution Summary

Two alternative solutions are provided for this combinatorics word problem. The first uses an intuitive approach. The second uses the classic 'm choose n' formula as the first step in solving the problem. Both detailed solutions arrive at the same answer.