In a study of 38% of adults questioned reported that their health was excellent. a researcher wishes to study the health of people living close to a nuclear power plant. find the probability that when 11 adults are randomly selected, at most 3 are in excellent health

This is a problem on binomial probability. We are looking for [tex]P\left(X\le 3\right)[/tex] for n=11, p=0.38, q=1-0.38=0.62.

We have
     [tex]P\left(X\le 3\right)=\sum _{i=0}^3\left(C\left(11,i\right)\cdot \left(0.38\right)^i\cdot \left(0.62\right)^{11-i}\right)[/tex]

     [tex]P\left(X\le 3\right)=C\left(11,0\right)\left(0.38\right)^0\left(0.62\right)^{11}+C\left(11,1\right)\left(0.38\right)^1\left(0.62\right)^{10}+C\left(11,2\right)\left(0.38\right)^2\left(0.62\right)^9+C\left(11,3\right)\left(0.38\right)^3\left(0.62\right)^8[/tex]

     [tex]P\left(X\le 3\right)=0.3455[/tex]