A biology instructor gives a quiz consisiting of truefalse i

A biology instructor gives a quiz consisiting of true-false items. She announces that the grades will be assigned as follows: A (14 or 15 correct); B (12 or 13 correct); C (10 or 11 correct); D (9 correct); F (8 or fewer). If a student has not studied for the quiz and just randomly guesses on each question, what is the probability that he or she will get a grade of....

a) A

b) B

c) C

d) D or F

e) C or better

Solution

P(correct answer) = p = 0.5
q = 1-p = 1-0.5 = 0.5
n = 15
P(r) = nCr*q^(n-r)*p^r
a)
Probability of getting grade A = P(r=14)+P(r=15)
P( X = 15 ) = ( 15 15 ) * ( 0.5^15) * ( 1 - 0.5 )^0
= 0.000031
P( X = 14 ) = ( 15 14 ) * ( 0.5^14) * ( 1 - 0.5 )^1
= 0.000458

P( X = 15 ) + P( X = 14 )= 0.000458 + 0.000031
= 0.000489

b) Probability of getting grade B = P(r=12)+P(r=13)
P( X = 12 ) = ( 15 12 ) * ( 0.5^12) * ( 1 - 0.5 )^3
= 0.013885
P( X = 13 ) = ( 15 13 ) * ( 0.5^13) * ( 1 - 0.5 )^2
= 0.003204

P( X = 12 ) + P( X = 13 ) = 0.013885 + 0.003204
= 0.017089

c)
Probability of getting grade C = P(r=10)+P(r=11)
P( X = 10 ) = ( 15 10 ) * ( 0.5^10) * ( 1 - 0.5 )^5
= 0.091644
P( X = 11 ) = ( 15 11 ) * ( 0.5^11) * ( 1 - 0.5 )^4
= 0.041656

P( X = 10 ) + P( X = 11 ) = 0.091644 + 0.041656 = 0.1333

d) Probability of getting a grade of D
P(D) =
P( X = 9 ) = ( 15 9 ) * ( 0.5^9) * ( 1 - 0.5 )^6
= 0.15274
or
P(F) =
P( X < = 8) = P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 15 8 ) * 0.5^8 * ( 1- 0.5 ) ^7 + ( 15 7 ) * 0.5^7 * ( 1- 0.5 ) ^8 + ( 15 6 ) * 0.5^6 * ( 1- 0.5 ) ^9 + ( 15 5 ) * 0.5^5 * ( 1- 0.5 ) ^10 + ( 15 4 ) * 0.5^4 * ( 1- 0.5 ) ^11 + ( 15 3 ) * 0.5^3 * ( 1- 0.5 ) ^12 + ( 15 2 ) * 0.5^2 * ( 1- 0.5 ) ^13 + ( 15 1 ) * 0.5^1 * ( 1- 0.5 ) ^14 + ( 15 0 ) * 0.5^0 * ( 1- 0.5 ) ^15
= 0.696381

P(D or F) = 0.15274 + 0.696381 = 0.849121

e) P(C or better) = P(C) + P(D) + P(F) = 0.1333 + 0.15274 + 0.696381 = 0.982421