Answer to Question #323693 in Statistics and Probability for Klum

Question #323693

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.


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Expert's answer
2022-04-07T16:38:25-0400

Xnumber  of  correct  answersp=0.1  correct  answerXBin(20,0.1)EX=mp=200.1=2Var(X)=mp(1p)=200.10.9=1.8Central  limit  theorem:XnEXnVar(X)N(0,1)P(X10)=P(X2201.8102201.8)==P(Z1.33333)=Φ(1.33333)=0.0912X-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


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