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Suppose that each of Barbara’s shots hits a wooden duck target with probability p₁, while each shot of Dianne’s hits it with probability p₂.

If the wooden duck is knocked over then the probability that both shots hit the duck is given by

$=\frac{P\left[\text{Both hits}\right]}{P\left[\text{at least one hits}\right]}$

$=\frac{P\left[\text{Both hits}\right]}{1−P\left[\text{no hits}\right]}$

$=\frac{p_1{p}_2}{1−q_1{q}_2}$

The probability that both shots hit the duck will be$\frac{p_1{p}_2}{1−q_1{q}_2}$.

Barbara and Dianne go target shooting. Suppose that each of Barbara’s shots hits a wooden duck target with probability p₁, while each shot of Dianne’s hits it with probability p₂.

If the wooden duck has knocked over then the probability that Barbara's shot struck the duck is shown below,

$P\left[\text{Barbara shot hit the duck |at least one hit(the wooden duck is knocked over)}\right]$

$=\frac{P\left[\text{Barbar hits}\right]}{P\left[\text{at least one hits}\right]}$

$=\frac{P\left[\text{Barbara hits}\right]}{1−P\left[\text{no hits}\right]}$

$=\frac{p_1}{1−q_1{q}_2}$

The probability of Barbara’s shot hit the duck is$=\frac{p_1}{1−q_1{q}_2}$.