Question

on m Assume that random guesses are made for 10 multiple choice questions SAT test. Each Each question has four choices Conly one is correct). Find the following probability values. a) p L Exactly 8 correct answers) 6) p ( At least 8 correct answers) c) p ( No correct answers) d) pl No more than one ca correct answer

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Answer 1

Here

Number of trials = n = 10

Probability of success = p = 1/4 = 0.25

The probability mass function of binomial distribution is,

, x = 0,1,2,...,n

n-1 p(X = x) = (2) pº(1 – p)

a)

$P(X=8)=\binom{10}{8}0.25^{8}(1-0.25)^{10-8}$

$\approx 0.00038624$

P( excatly 8 correct answers) = 0.00038624
000039

b)

$P(X\geq 8)=P(X=8)+P(X=9)+P(X=10)$

$=\binom{10}{8}0.25^{8}(1-0.25)^{10-8}+\binom{10}{9}0.25^{9}(1-0.25)^{10-9}$

$+\binom{10}{10}0.25^{10}(1-0.25)^{10-10}$

$=0.00038624+0.00002861+0.00000095$

= 0.0004158

P( at least 8 correct answers) = 0.0004158 $\approx 0.0004$

c)

$P(X=0)=\binom{10}{0}0.25^0(1-0.25)^{10-0}$

$=1*1*0.75^1^0$

  $\approx 0.0563135$

P( No correct answers) = 0.0563135  $\approx 0.0563$

d)

$P(X\leq 1)=P(X=0)+P(X=1)$

$=\binom{10}{0}0.25^0(1-0.25)^{10-0}+\binom{10}{1}0.25^1(1-0.25)^{10-1}$

$=0.0563135+0.1877117$

= 0.2440252

P( No more than on e correct answer) =0.2440252 $\approx 0.2440$