Answer
$P(at~least~8~will~not~die)=0.9885$
Work Step by Step
If at least 8 will not die, no more than 2 will die
$P(at~least~8~will~not~die)=P(no~more~than~2~will~die)=P(X\leq2)=P(0)+P(1)+P(2)={}_{10}C_{0}~0.05^0~(0.95)^{10}+{}_{10}C_{1}~0.05^1~(0.95)^{9}+{}_{10}C_{2}~0.05^2~(0.95)^{8}=\frac{10!}{0!\times(10-0)!}\times1\times0.95^{10}+\frac{10!}{1!\times(10-1)!}\times0.05\times(0.95)^{9}+\frac{10!}{2!\times(10-2)!}\times0.0025\times(0.95)^{8}=1\times1\times0.95^{10}+10\times0.05\times(0.95)^{9}+45\times0.0025\times(0.95)^{8}=0.9885$
