Answer to Question #330177 in Statistics and Probability for Robera

Question

The probability that a freshman entering AAU (Science Faculty) will survive first semester is 0.92. (From

1993/94 academic year statistics). Assuming this pattern remain unchanged over the subsequent years, what

is the probability that among 100 randomly selected freshmen in first semester,

a) None will survive?

b) Exactly 97 will survive?

c) At least three will survive?

Answer 1

$p=0.92\\X\sim Bin\left( 100,0.92 \right) \\a:\\P\left( X=0 \right) =\left( 1-0.92 \right) ^{100}=2.03704\times 10^{-110}\\b:\\P\left( X=97 \right) =C_{100}^{97}\cdot 0.92^{97}\cdot \left( 1-0.92 \right) ^3=0.0254331\\c:\\P\left( X\geqslant 3 \right) =1-P\left( X=0 \right) -P\left( X=1 \right) -P\left( X=2 \right) =\\=1-2.03704\times 10^{-110}-C_{100}^{1}\cdot 0.92^1\cdot \left( 1-0.92 \right) ^{99}-C_{100}^{99}\cdot 0.92^2\cdot \left( 1-0.92 \right) ^{98}=\\=1-2.92804\times 10^{-106}$