A student takes a 20-question, true/false exam and guessed on each question. Find the probability of passing if the lowest passing grade is 15 correct out of 20.

What you want to know is the chance probability of obtaining 15, 16, 17, 18, 19, or 20 correct.

Prob. of 15 = .5^15...
Prob. of 20 = .5^20

Either-or probability = sum of individual probabilities.

To find the probability of passing the exam, we need to calculate the probability of getting at least 15 correct answers out of 20, given that the student guessed on each question.

First, let's calculate the probability of getting exactly 15 answers correct out of 20.

The probability of guessing the correct answer on any given question is 1/2 since it is a true/false exam. Therefore, the probability of guessing 15 answers correctly is (1/2)^15.

Next, let's calculate the probability of getting 16, 17, 18, 19, or 20 answers correct out of 20.

The probability of getting exactly 16 correct answers out of 20 is (1/2)^16.

The probability of getting exactly 17 correct answers out of 20 is (1/2)^17.

Similarly, the probability of getting exactly 18, 19, and 20 correct answers out of 20 is (1/2)^18, (1/2)^19, and (1/2)^20, respectively.

Finally, we need to sum up all these probabilities to find the probability of passing:

P(Passing) = P(15 correct) + P(16 correct) + P(17 correct) + P(18 correct) + P(19 correct) + P(20 correct)
= (1/2)^15 + (1/2)^16 + (1/2)^17 + (1/2)^18 + (1/2)^19 + (1/2)^20

Calculating this sum will give us the desired probability.

To find the probability of passing, we need to calculate the probability of getting at least 15 correct answers out of 20, given that the student guessed on each question.

Let's break it down step-by-step:

Step 1: Calculate the probability of getting exactly 15 correct answers.
To solve this, we use the binomial probability formula:
P(X = k) = (nCk) * p^k * (1 - p)^(n - k)

In this case, n = 20 (total number of questions), k = 15 (desired number of correct answers), and p = 0.5 (the probability of guessing correctly on a true/false question).

Using the binomial coefficient (nCk), which represents the number of ways to choose k items from a set of n items, we have:
(20C15) * (0.5)^15 * (0.5)^(20 - 15)

Step 2: Calculate the probability of getting exactly 16, 17, 18, 19, and 20 correct answers.
Using the same formula from step 1, we calculate each probability individually:
P(X = 16) = (20C16) * (0.5)^16 * (0.5)^(20 - 16)
P(X = 17) = (20C17) * (0.5)^17 * (0.5)^(20 - 17)
P(X = 18) = (20C18) * (0.5)^18 * (0.5)^(20 - 18)
P(X = 19) = (20C19) * (0.5)^19 * (0.5)^(20 - 19)
P(X = 20) = (20C20) * (0.5)^20 * (0.5)^(20 - 20)

Step 3: Sum up the probabilities from step 1 and step 2.
Since we want to find the probability of passing, we need to sum up the probabilities of getting at least 15 correct answers:
P(passing) = P(X >= 15) = P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20)

Once you calculate each individual probability in steps 1 and 2, add them up to find the overall probability of passing.