A student is writing a multiple choice test consisting of 40 questions, each of which provides four possible choices. He is certain that he has 16 questions correct. If he guesses for all of the remaining 24 questions, what is the probability that he will pass the test?
Assuming that a pass is getting at least 20 of the 40 correct ....
What the student CANNOT have is
zero of the remaining 24 correct : C(24,0) (1/4)^0 (3/4)^24 = .001003391
1 of the remaining 24 correct : C(24,1) (1/4) (3/4)^23 = .00802713
2 of the remaining 24 correct : C(24,2) (1/4)^2 (3/4)^22 = .030770665
3 of the remaining 24 correct : C(24,3) (1/4)^3 (3/4)^21 = .075217183
prob he will pass = 1 - (sum of the above 4 cases)
= ...
check my arithmetic
Assuming:
1. he got indeed the 16 questions correctly.
2. the remaining 24 answers were random choices
3. there was exactly one correct answer
4. passing grade is 50%
to each of the remaining questions
then he will need 4 more correct answers to get 50%.
Let
p=probability of guessing a question correctly = 0.25
q=1-p=probability of guessing incorrectly = 0.75
The probability of getting 0 to 3 correct guesses (i.e. fail) is
C(24,0)*q^24*p*0 +
C(24,1)*q^23*p*1 +
C(24,2)*q^22*p*2 +
C(24,3)*q^21*p*3
=0.0010+0.0080+0.00308+0.00752
=0.1150
where C(n,r)=n!/((n-r)!r!)
Therefore the probability that he will pass (at 50%) is
1-0.1150=0.8850
Here we go again!
What is the probability of that ? lol
To determine the probability that the student will pass the test, we need to calculate the probability of getting a passing score.
Since a passing score is not defined in the question, we will assume that the student needs to get at least 60% of the questions correct to pass. With 40 questions on the test, a passing score would be 24 or more correct answers.
We know that the student has already answered 16 questions correctly, so we need to calculate the probability of getting 8 or more additional answers correct out of the 24 remaining questions.
To calculate the probability, we will use the binomial probability formula:
P(X ≥ k) = 1 - P(X < k), where X is a binomial random variable, k is the number of correct answers needed, and P(X < k) is the cumulative probability of getting less than k correct answers.
To calculate the cumulative probability, we will use the binomial cumulative distribution function (CDF). The binomial CDF calculates the probability of getting up to a certain number, k, of correct answers.
The formula for the binomial CDF is:
P(X < k) = ∑[i=0 to k-1] (nCi) * p^i * (1-p)^(n-i), where n is the total number of trials (remaining questions), p is the probability of success (1 out of 4 choices), and i is the number of successes (correct answers).
Now, let's calculate the probability of passing the test.