3. The problem consists of a brief introduction to discrete probability. A multi

Question

3. The problem consists of a brief introduction to discrete probability. A multiple choice test consists of five questions, each of which has four choices. Each question has only one correct answer.

a. How many ways are there to fill out an answer sheet?

b. How many ways of the possible answer sheets have exactly four correct responses?

c. The probability of getting exactly four correct is determine by dividing the number of ways in which one can get four correct divided by the number of ways in which the answer sheet can be filled out. What is the probability of getting four correct?

d. What is the probability of answering all 5 correctly?

e. What is the probability of getting at least 3 correct?

Explanation / Answer

Solution:

(a)

Number of ways are there to fill out an answer sheet 5 question with choices

Let first can be chosen as = 4 ways

Similerly second = 4 ways and so on

therefore for five question the answer sheet can be fill out = 4 x 4 x4x4x4

= 45 = 1024 ways

(b) the possible answer sheets have exactly four correct responses can be determined by

one correct choice out of out of 4 option of 4 question while remaining 5th question's answer can be 3 ways to make it wrong

for correct choce we have only one way to chose it

therefore = 1 x 1 x 1 x 1 x 3

( they choice is independent )

the no. of ways of the possible answer sheets have exactly four correct responses = 4 ways

(c)

the probability of getting four correct = (the no. of ways of the possible answer sheets have exactly four correct responses) / or five question the answer sheet can be fill out

= 3/45 =3/1024 = 0.00292968

(d) probability of answering all 5 correctly

the no. of ways of the possible answer sheets have exactly 5 correct = 1x1x1x1x1 = 1

probability of answering all 5 correctly = 1/45 = 1/1024

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