SOLUTION: A multiple-choice test consists of 23 questions with possible answers of a, b, c, d, e. Estimate the probability that with random guessing, the number of correct answers is at leas

Hi, there--

The Problem:
A multiple-choice test consists of 23 questions with possible answers of a, b, c, d, e. Estimate the 
probability that with random guessing, the number of correct answers is at least 8.

Solution:
This situation can be modeled by the Binomial Distribution because
1: The number of observations n is fixed :: 23 questions on the M/C test.
2: Each observation is independent :: each question is independent of the others.
3: Each observation represents one of two outcomes :: correct answer or incorrect answer.
4: The probability of "success" p is the same for each outcome :: 5 answer choices, 1 correct, 4 incorrect.

The formula for the Binomial Distribution is 

P(X = k) = [nCk] * [p^k] * [q^(n-k)]

What do all these variables mean?

For choosing exactly 8 correct answers: 
X = the random variable representing the event of randomly guessing a correct answer
n = the number of trials (questions on the test) = 23
k = the number of successful trials (correct answers) = 8
n-k = the number of unsuccessful trials (wrong answers) = 23-8 = 15
p = the probability of success, 5 choices (a b c d e) and one is correct = 1/5 = 0.2
q = the probability of failure, 5 choices, and 4 are wrong = 4/5 = 0.8

nCk = the combination "n choose k," how many ways to choose k items from a set of n items = 23C8
nCK = n!/(k!(n-k)!) = 23!/(8!15!)
P(X = k) = the probability that with random guessing there will be exactly 8 right answers.

Your problem asks for the probability that there will be at least 8 correct answers, P(k≥8)

P(k≥8) = P(k=8) + P(k=9) + P(k=10) + ... + P(k=22) + P(k=23)

This is a lots of calculating. We can use the fact that the sum of all possible probabilities in a situation 
must equal 1 to simplify our work a little.

P(k≥8) = 1 - P(k<8)

or

P(k≥8) = 1 - [ P(k=0) + P(k=1) + P(k=2) + P(k=3) + P(k=4) + P(k=5) + P(k=6) + P(k=7) ]

I'll calculate a few of these for you:

0 correct answers:
P(k=0) = [23C0] * [(0.8)^23] = [1] = 0.0059

1 correct answer:
P(k=1) = [23C1] * [(0.8)^22] = [23] = 0.0339

Calculate each probability P(k=0) through P(k=7) and subtract their sum from 1.

If you have questions about this, or you want to check your answer, feel free to email me.

Mrs. Figgy
math.in.the.vortex@gmail.com