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Independent alarms

ProbabilityEasy

A risk system has $2$ independent alarms. On a given day the first fires with probability $0.3$ and the second with probability $0.5$ . What is the probability that at least one of them fires? Give a decimal.

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Hint

Look at the complement: no alarm fires.

Solution

The first alarm stays silent with probability $1 - 0.3 = 0.7$ .
The second alarm stays silent with probability $1 - 0.5 = 0.5$ .
By independence, both stay silent with probability $0.7 \times 0.5 = 0.35$ .
So at least one fires with probability $1 - 0.35 = 0.65$ .

Answer: $0.65$

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