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I read the following problem and I cannot get the right solution:

We consider a system of 12 sensors. The probability of one sensor to detect a signal with a magnitude higher than a is $p=\frac{1}{2\pi }\int_{x=a}^\infty \exp(\frac{-x^2}{2})$. What is the joint probability that 5 sensors detect this signal? It is stated that if a=2.8, the five signals yield to a joint probability corresponding to 5.7


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Use the binomial distribution. The probability that exactly 5 sensors of 12 will register is $\binom {12}{5}p^5(1-p)^7$. Just sum up the chance that 5, 6, ... 12 will register. The claim (which I have not verified) is that the sum over 5, 6, ... 12 out of 12 for a=2.8 equals the chance of a single unit at a=5.7

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