Four groups can be formed from N shooters. a1 are excellent shooters a2 are good ones a3 are fair ones and a4 are poor. The probability that a shooter in group i hits the target is pi. Two shooters are chosen at random to shoot same target. Find the probability that target will be hit at least once.
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$\begingroup$ Do we have $a_1+a_2+a_3+a_4=N$ here? And what are your own thoughts about it? Add this to your question. $\endgroup$– drhabOct 16, 2016 at 11:30
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1 Answer
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If the first shooter chosen comes from group $i$ and the second comes from group $j$ then the probability of at least one hit is: $$1-(1-p_i)(1-p_j)$$
Denoting the probability that the first shooter chosen comes from group $i$ and the second comes from group $j$ by $p_{ij}$ we can express the probability of at least one hit by:$$\sum_{i=1}^4\sum_{j=1}^4p_{ij}[1-(1-p_i)(1-p_j)]$$
It remains to find expressions for $p_{ij}$ in $a_1,a_2,a_3,a_4$ which I will leave up to you.
Also note that $p_{ij}=p_{ji}$
