A student took a 25 question true and false test answering the question randomly without reading. If a score of 18 is correct is the minimum passing graade, how likely is it that the random guesser will pass the test?

This is binomial although there is a slight twist in the Q

You can use a binomial probability table to determine the probability. Find P(18) through P(25), then add all the probabilities together for a total.

38

To find out the likelihood of a random guesser passing the test, we need to use the binomial distribution. In this case, the random guesser has a 50% chance of guessing correctly on each question (since it is a true/false test).

The probability of passing the test with a minimum passing grade of 18 out of 25 questions can be calculated by summing the probabilities of getting 18, 19, 20, 21, 22, 23, 24, or 25 questions correct.

To calculate the probability of getting exactly k questions correct out of 25, we can use the binomial probability formula:

P(X = k) = (nCr) * (p^k) * (q^(n-k)),

Where:
- n is the total number of questions (25 in this case),
- k is the number of questions answered correctly,
- p is the probability of answering a question correctly (0.5 in this case),
- q is the probability of answering a question incorrectly (1 - p = 0.5), and
- nCr is the combination formula or the number of ways to choose k items from a set of n items.

Now, let's calculate the probability of passing the test with a minimum passing grade of 18:

P(X ≥ 18) = P(X = 18) + P(X = 19) + P(X = 20) + ... + P(X = 25).

P(X = 18) = (25C18) * (0.5^18) * (0.5^(25-18))
P(X = 19) = (25C19) * (0.5^19) * (0.5^(25-19))
P(X = 20) = (25C20) * (0.5^20) * (0.5^(25-20))
...
P(X = 25) = (25C25) * (0.5^25) * (0.5^(25-25))

To calculate each of these probabilities, we need the combination formula:

nCr = n! / (r!(n-r)!)

Let's calculate the probabilities and sum them up to find the overall probability of passing the test.