To solve this problem, we need to understand that it's a binomial probability problem. The binomial probability formula is P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k)), where:

- P(X=k) is the probability of k successes in n trials
- C(n, k) is the combination of n items taken k at a time
- p is the probability of success
- n is the number of trials

In this case, the probability of success (p) is 0.5 (since the student is guessing and there are only two possible outcomes - true or false), and the number of trials (n) is 3 (since there are three questions).

We want to find the probability that the student gets at least 2 questions correct, which means they could get 2 or 3 questions correct. So we need to find P(X=2) and P(X=3) and add them together.

First, let's find P(X=2):

- C(3, 2) = 3 (there are 3 ways to choose 2 successes from 3 trials)
- p^2 = 0.5^2 = 0.25 (the probability of 2 successes)
- (1-p)^(n-2) = (1-0.5)^(3-2) = 0.5 (the probability of 1 failure)

So P(X=2) = 3 * 0.25 * 0.5 = 0.375

Next, let's find P(X=3):

- C(3, 3) = 1 (there is 1 way to choose 3 successes from 3 trials)
- p^3 = 0.5^3 = 0.125 (the probability of 3 successes)
- (1-p)^(n-3) = (1-0.5)^(3-3) = 1 (the probability of 0 failures)

So P(X=3) = 1 * 0.125 * 1 = 0.125

Finally, add P(X=2) and P(X=3) together to find the probability of getting at least 2 questions correct:

P(X=2) = P(X=2) + P(X=3) = 0.375 + 0.125 = 0.50 or 50% when expressed as a percentage.

Question

To solve this problem, we need to understand that it's a binomial probability problem. The binomial probability formula is P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k)), where:

- P(X=k) is the probability of k successes in n trials
- C(n, k) is the combination of n items taken k at a time
- p is the probability of success
- n is the number of trials

In this case, the probability of success (p) is 0.5 (since the student is guessing and there are only two possible outcomes - true or false), and the number of trials (n) is 3 (since there are three questions).

We want to find the probability that the student gets at least 2 questions correct, which means they could get 2 or 3 questions correct. So we need to find P(X=2) and P(X=3) and add them together.

First, let's find P(X=2):

- C(3, 2) = 3 (there are 3 ways to choose 2 successes from 3 trials)
- p^2 = 0.5^2 = 0.25 (the probability of 2 successes)
- (1-p)^(n-2) = (1-0.5)^(3-2) = 0.5 (the probability of 1 failure)

So P(X=2) = 3 * 0.25 * 0.5 = 0.375

Next, let's find P(X=3):

- C(3, 3) = 1 (there is 1 way to choose 3 successes from 3 trials)
- p^3 = 0.5^3 = 0.125 (the probability of 3 successes)
- (1-p)^(n-3) = (1-0.5)^(3-3) = 1 (the probability of 0 failures)

So P(X=3) = 1 * 0.125 * 1 = 0.125

Finally, add P(X=2) and P(X=3) together to find the probability of getting at least 2 questions correct:

P(X>=2) = P(X=2) + P(X=3) = 0.375 + 0.125 = 0.50 or 50% when expressed as a percentage.

Knowee AI · Accepted Answer