SOLUTION: A quiz consists of 6 multiple-choice questions. Each question has 4 possible answers. A student is unprepared, and he has no choice but guessing answers completely at random. He

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Question 1051952: A quiz consists of 6 multiple-choice questions. Each question has 4 possible answers. A
student is unprepared, and he has no choice but guessing answers completely at random.
He passes the quiz if he gets at least 3 questions correctly. What is the probability that he
will pass?

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
The probability that he will guess an answer right is 1/4.
The probability that he will guess an answer wrong is 3/4.

This is a cumulative binomial probability where the number of trials 
is n=6, with the probability of a success in 1 trial is 1/4.

With a TI-83 or 84,  it's 1-binomcdf(6,1/4,2). Ans: 0.1694335937

Or without a calculator that has binomial probability feature:

Case 1: Guessing exactly 3 correct:

Choosing exactly 3 to get correct = Choosing exactly
3 to get wrong = 6C3 = (6*5*4)/(3*2*1) = 20 ways.
Guess the wrong answer for the first wrong one 3 ways.
Guess the wrong answer for the second wrong one 3 ways.
Guess the wrong answer for the third wrong one 3 ways.
That's 20×3×3×3 = 540 ways

Case 2: Guessing exactly 4 correct:

Choosing exactly 4 to get correct = Choosing exactly
2 to get wrong = 6C2 = (6*5)/(2*1) = 15 ways.
Guess the wrong answer for the first wrong one 3 ways.
Guess the wrong answer for the second wrong one 3 ways.
That's 15×3×3 = 135 ways

Case 3: Guessing exactly 5 correct:

Choosing exactly 5 to get correct = Choosing exactly
1 wrong = 6C1 = 6 ways.
Guess the wrong answer for the wrong one 3 ways.
That's 6×3 = 18 ways

Case 4: Guessing all 6 correct
There is only 1 way to guess them all correct

Total number of ways to pass: 540+135+18+1 = 694 ways to pass

Total number of ways to answer:

(4 ways to answer question 1) times
(4 ways to answer question 2) times
(4 ways to answer question 3) times
(4 ways to answer question 4) times
(4 ways to answer question 5) times
(4 ways to answer question 6) equals 4×4×4×4×4×4 = 46 = 4096

matrix%281%2C3%2CProbability%2Cof%2Cpassing%29%22%22=%22%22694%2F4096%22%22=%22%22347%2F2048%22%22=%22%220.1694335938 
  
Edwin