A biology quiz consists of twelve multiple-choice questions. Eight must be answered correctly to receive a passing grade. If each question has five possible answers, of which only one is correct, what is the probability that a student who guesses at random on each question will pass the examination? (Round your answer to four decimal places.)
a multiple choice test has 30 questions and each has five possible answers, of which one is correct, if a student guesses on every question, find the probability of fetting exactly 12 correct. state why is this a binomial random variable
To find the probability that a student who guesses at random on each question will pass the examination, we need to determine the number of ways in which they can answer at least eight questions correctly.
Since each question has five possible answers, the probability of guessing the correct answer to any specific question is 1/5. Similarly, the probability of guessing the incorrect answer is 4/5.
To calculate the probability of getting exactly eight questions correct out of twelve, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * q^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes
- C(n, k) is the number of combinations of n items taken k at a time
- p is the probability of success in any individual trial
- q is the probability of failure in any individual trial
- n is the total number of trials
In this case, n = 12 (total number of questions), k = 8 (number of correct answers needed to pass), p = 1/5 (probability of guessing correctly), and q = 4/5 (probability of guessing incorrectly).
Using the formula, the probability of getting exactly eight questions correct is:
P(X = 8) = C(12, 8) * (1/5)^8 * (4/5)^(12-8)
Calculating this expression:
P(X = 8) = 495 * (1/5)^8 * (4/5)^4
Now, to get the probability of passing the examination, we need to consider all the possible numbers of correct answers that can lead to passing. Since a passing grade is achieved by answering at least eight questions correctly, we need to calculate the probabilities for getting exactly 8, 9, 10, 11, and 12 questions correct.
To calculate the probabilities for each scenario and then sum them up:
P(pass) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)
P(pass) = [C(12, 8) * (1/5)^8 * (4/5)^4] + [C(12, 9) * (1/5)^9 * (4/5)^3] + [C(12, 10) * (1/5)^10 * (4/5)^2] + [C(12, 11) * (1/5)^11 * (4/5)^1] + [C(12, 12) * (1/5)^12 * (4/5)^0]
Calculating this expression will give us the probability that a student who guesses at random on each question will pass the examination.