Question

The multiple-choice question on a combinatorics quiz contains 20 questions with ten possible answers each (where only 1 answer is true). Compute the probability of randomly guessing the correct answers and getting at least 10 correct answers to 20 questions (Event A). Find the mathematical expectation and variance of the number of correct answers.

Answer 1

$X-number\,\,of\,\,correct\,\,answers\\p=0.1 -\,\,correct\,\,answer\\X\sim Bin\left( 20,0.1 \right) \\EX=mp=20\cdot 0.1=2\\Var\left( X \right) =mp\left( 1-p \right) =20\cdot 0.1\cdot 0.9=1.8\\Central\,\,\lim\mathrm{i}t\,\,theorem:\\\frac{X-nEX}{\sqrt{nVar\left( X \right)}}\sim N\left( 0,1 \right) \\P\left( X\geqslant 10 \right) =P\left( \frac{X-2}{\sqrt{20\cdot 1.8}}\geqslant \frac{10-2}{\sqrt{20\cdot 1.8}} \right) =\\=P\left( Z\geqslant 1.33333 \right) =\varPhi \left( -1.33333 \right) =0.0912$