44.Richard has just been given a 10-question multiple-choice quiz in his history

Question

44.Richard has just been given a 10-question multiple-choice quiz in his history class. Each question has five answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all ten questions, find the indicated probabilities. (Round your answers to three decimal places.)

(a) What is the probability that he will answer all questions correctly?


(b) What is the probability that he will answer all questions incorrectly?


(c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table.


Then use the fact that P(r 1) = 1 P(r = 0).


Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference?

They should be equal, but may not be due to table error.They should be equal, but may differ slightly due to rounding error.    They should be equal, but differ substantially.They should not be equal, but are equal.

(d) What is the probability that Richard will answer at least half the questions correctly?

Explanation / Answer

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

a.
P( X = 10 ) = ( 10 10 ) * ( 0.2^10) * ( 1 - 0.2 )^0
= 0.000000102

b.
P( X < 1) = P(X=0)   
= ( 10 0 ) * 0.2^0 * ( 1- 0.2 ) ^10 +
= 0.1074
P( X > = 1 ) = 1 - P( X < 1) = 0.8926

c.
P( X > = 1 ) = 1 - P( X < 1) = 0.8926

d.
P( X < 5) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 10 4 ) * 0.2^4 * ( 1- 0.2 ) ^6 + ( 10 3 ) * 0.2^3 * ( 1- 0.2 ) ^7 + ( 10 2 ) * 0.2^2 * ( 1- 0.2 ) ^8 + ( 10 1 ) * 0.2^1 * ( 1- 0.2 ) ^9 + ( 10 0 ) * 0.2^0 * ( 1- 0.2 ) ^10
= 0.9672

P( X > = 5 ) = 1 - P( X < 5) = 0.0328

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