Question

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

please show work.

Answer 1

Sol)

Given

n = 10

p = 1/4 = 0.25

q = 1 - p = 1 - 0.25 = 0.75

a)

Exactly 8 are correct

P( X = r ) = nCr * ( p ) ^ r * ( q ) ^ ( n - r )

P( X = 8 ) = 10C8 * ( 0.25 ) ^ 8 * ( 0.75 ) ^ ( 10 - 8 )

= 45 * ( 0.25 ) ^ 8 * ( 0.75 ) ^ 2

= 0.0003862

b)

Atleast 8 are correct

P( X >= 8 ) = P( X = 8 ) + P( X = 9 ) + P( X = 10 )

P( X = 9 ) = 10C9 * ( 0.25 ) ^ 9 * ( 0.75 ) ^ ( 10 - 9 )

= 10 * (0.25 ) ^ 9 * ( 0.75 ) ^ 1

= 0.0000286

P( X = 10 ) = 10C10 * ( 0.25 ) ^ 10 * ( 0.75 ) ^ ( 10 - 10)

= 1 * ( 0.25 ) ^ 10

= 0.00000001

P( X >= 8 ) = P( X = 8 ) + P( X = 9 ) + P( X = 10 )

= 0.0003862 + 0.0000286 + 0.000000

= 0.000415802

c)

Probability that none are correct

P( X = 0 ) = 10C0 * ( 0.25 ) ^ 0 * ( 0.75 ) ^ ( 10 - 0)

= 1 * ( 0.75 ) ^ 10

= 0.0563135

d)

P( no more than one correct answer )

P( X <= 1 ) = P( X = 0 ) + P( X = 1 )

P( X = 1 ) = 10C1 * ( 0.25 ) ^ 1 * (0.75 ) ^ ( 10 - 1 )

= 10 * ( 0.25 ) * ( 0.75 ) ^ 9

= 0.1877117

P( X <= 1 ) = P( X = 0 ) + P( X = 1 )

= 0.0563135 + 0.1877117

= 0.2440252