A professor gives a test with 100 true-false questions. If 60 or more are necessary to pass, what is the probability that a student will pass by random guessing?

Here's one way to do this problem; there may be others. Using the normal approximation to the binomial distribution, we have the following:

p = .5
q = .5 --> q = 1 - p
x = 60
n = 100

We now need to find mean and standard deviation.
mean = np = (100)(.5) = 50
standard deviation = √npq = √(100)(.5)(.5) = √(25) = 5

Using z-scores:
z = (x - mean)/sd
z = (60 - 50)/5 = 2

Use a normal distribution table to find the probability. If the table shows from mean to z, check from mean to z = 2.00. Then take the value you find in the table and subtract it from .5000 for your probability.

I hope this will help.

To find the probability of passing by random guessing, we need to determine the number of questions the student needs to answer correctly in order to pass.

Since the test has 100 true-false questions, there are two possible outcomes for each question (true or false). Therefore, the probability of guessing a question correctly is 1/2 or 0.5.

To pass the test, the student needs to answer 60 or more questions correctly. Let's calculate the probability of getting each possible score from 60 to 100.

To get exactly 60 correct answers:
- The probability of getting a question right is 0.5, so the probability of guessing 60 questions correctly is (0.5)^60.

To get exactly 61 correct answers:
- The probability of getting the one question wrong is 0.5, so the probability of guessing 61 questions correctly is (0.5)^61.

We follow similar logic to calculate the probability of guessing 62 to 100 questions correctly.

Finally, to obtain the total probability of passing, we sum the probabilities of getting 60 or more questions correct:

P(passing) = P(getting 60 questions correct) + P(getting 61 questions correct) + ... + P(getting 100 questions correct).

In other words, we add up the probabilities of each individual score.

Calculating this would require summing 41 terms, so it would be quite tedious to do manually. However, you can use a calculator or a computer to simplify the calculation.

Alternatively, you could use a binomial probability formula to calculate the probability of passing. The binomial formula allows us to calculate the probability of getting a certain number of successes (correct guesses) in a given number of trials (questions).

For this case, we have 100 trials (questions), with the probability of success (getting a question right) being 0.5, and we want to find the probability of getting 60 or more successes.

Using a binomial probability calculator, you can input these values to determine the probability of passing by random guessing.

In summary, you can either manually calculate the probabilities for each score and sum them up, or simplify the calculation by using a binomial probability formula or calculator.