16.02 Binomial distributions

Find the value of $\nCr{5}{3}\times\left(0.8\right)^3\times\left(0.2\right)^2+\nCr{5}{4}\times\left(0.8\right)^4\times0.2+\nCr{5}{5}\times\left(0.8\right)^5\times\left(0.2\right)^0$5C3×(0.8)3×(0.2)2+5C4×(0.8)4×0.2+5C5×(0.8)5×(0.2)0.

$X$X is a binomial variable with the probability mass function $P$P$($($X=k$X=k$)$)$=$=$\nCr{5}{k}\times\left(0.7\right)^k\times\left(0.3\right)^{5-k}$5Ck×(0.7)k×(0.3)5−k.

Census data show that $80%$80% of the population in a particular country have brown eyes.

A random sample of $900$900 people is selected from the population.