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A storeroom has two smoke detectors that work independently of each other…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A storeroom has two smoke detectors that work independently of each other. In a fire, the first detector sounds with probability and the second with probability . Find the probability that both detectors sound, the probability that exactly one sounds and the probability that at least one sounds. Give decimals to three places.
Need a reminder? The key ideaShow
Events are independent when , which is the same as . Test the product, never assume it. For independent events, .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. Both sound. .
Step 2. Exactly one sounds. The first sounds and the second is silent: . The second sounds and the first is silent: . The sum is .
Step 3. At least one sounds. Both silent has probability . So the answer is .
Answer: , and .
Check. The three cases both, exactly one and neither add up: .
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