Mary took a 20 question multiple-choice exam where there are 4 choices for each question and only 1 of those choices is correct. Rather than reading the question, Mary simply puts a random choice of answer down for each question. Determine the probability that Mary gets exactly 8 of the 20 questions correct.
The problem satisfies the following conditions:
-the experiment is a Bernoulli experiment (i.e. each trial has one of two outcomes)
- the probability of each trial is known remains constant throughout the experiment
- each trial is independent of the others.
This indicates a binomial distribution.
For exactly 8 correct answers, we calculate as follows:
p=prob. for success (answer correct)
q=prob. for failure (answer incorrect)
= 1-p
n=number of trials (20)
r=number of successes (8)
The probability of exactly 8 successes out of 20 is given by
P(8)=C(20,8)p^8q^(20-8)
where (20,8) is the binomial coefficient for p^8 , where
P(n,r)=n!/(r!(n-r)!)
For P(8), I get about 6%, or 0,06
To determine the probability that Mary gets exactly 8 of the 20 questions correct, we need to use the concept of binomial probability.
In this case, Mary has 4 choices for each of the 20 questions. Out of the 4 choices, only 1 is correct, so the probability of Mary randomly guessing the correct answer for each question is 1/4.
Now, to calculate the probability of getting exactly 8 questions correct, we need to use the binomial probability formula:
P(X=k) = nCk * p^k * (1-p)^(n-k)
where:
- P(X=k) is the probability of getting exactly k successes (in this case, getting exactly 8 questions correct)
- n is the total number of trials (in this case, 20 questions)
- k is the desired number of successes (in this case, 8 questions)
- p is the probability of success on each trial (in this case, 1/4, as Mary randomly guesses the correct answer)
- (1-p) is the probability of failure on each trial
Now, let's substitute the values into the formula:
P(X=8) = 20C8 * (1/4)^8 * (1 - 1/4)^(20-8)
To calculate 20C8 (known as "20 choose 8"), we use the combination formula:
nCr = n! / (r! * (n-r)!)
P(20C8) = 20! / (8! * (20-8)!)
Now, let's calculate the values:
P(20C8) = 20! / (8! * 12!)
P(20C8) = 20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 / (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
P(20C8) = 125,970
Substituting the values back into the binomial probability formula:
P(X=8) = 125,970 * (1/4)^8 * (1 - 1/4)^(20-8)