Question

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Problem 7. Ten hunters are waiting for ducks to fly by. When a flock of ducks flies overhead. the hunters fire at the same time, but each chooses his target at random, independently of the others. If each hunter independently hits his target with probability p, compute the expected number of ducks that escape unhurt when a flock of size 10 flies overhead.

Answer 1

$Let~X_i=1~if~ith~duck~is~unhurt.\\ =0~otherwise.\\ P(X_i=1)=Probability~that~none~of~10~hunters~hit~ith~duck\\ =(1-p)^{10},~since~hits~are~independent\\ E(X_i)=P(X_i=1)=(1-p)^{10},~i=1(1)10\\ Let~Y=number~of~ducks~that~escape~unhurt=X_1+X_2+...+X_{10}\\ Expected~number~of~ducks~that~escape~unhurt=E(Y)\\=\sum_{i=1}^{10}E(X_i)=10(1-p)^{10}$