Richard has just been given a 6question multiplechoice quiz

Richard has just been given a 6-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 six 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. Then use the fact that P(r 1) = 1 P(r = 0).

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

Solution

The probability of getting a question correctly is p = 1/5 = 0.2.

a)

P(All correct) = (0.2)^6 = 0.000064 [answer]

b)

P(all incorrect) = (1-0.2)^6 = 0.262144 [answer]

c)

P(at least one correct) =1 - P(all incorrect) = 0.737856 [answer]

d)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    6      
p = the probability of a success =    0.2      
x = our critical value of successes =    3      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   2   ) =    0.90112
          
Thus, the probability of at least   3   successes is  
          
P(at least   3   ) =    0.09888 [ANSWER]