A student randomly guesses on 10 true/false questions. use the binomial model to determine the probability that the student gets 5 out of 10 questions right. Show all your steps.
The binomial model can be used to calculate the probability of getting a specific number of successes (in this case, 5) in a fixed number of independent trials (10), where each trial has only two possible outcomes (true or false).
The formula for the binomial probability is:
P(X = k) = C(n, k) * p^k * q^(n-k)
where:
- P(X = k) is the probability of getting exactly k successes
- n is the number of trials
- k is the number of successes
- p is the probability of success in a single trial
- q is the probability of failure in a single trial (q = 1 - p)
- C(n, k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials and can be calculated as C(n, k) = n! / (k!(n-k)!)
In this case, n = 10 (number of trials), k = 5 (number of successes), p = 1/2 (probability of success in a single trial), and q = 1 - p = 1/2 (probability of failure in a single trial).
Plugging these values into the formula:
P(X = 5) = C(10, 5) * (1/2)^5 * (1/2)^(10-5)
Calculating C(10, 5):
C(10, 5) = 10! / (5!(10-5)!) = 10! / (5! * 5!) = (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1) = 252
Plugging this into the formula:
P(X = 5) = 252 * (1/2)^5 * (1/2)^(10-5)
P(X = 5) = 252 * 1/32 * 1/32
P(X = 5) = 252 / (32 * 32)
P(X = 5) = 252 / 1024
Using a calculator or simplifying the fraction:
P(X = 5) ≈ 0.2461
Therefore, the probability that the student gets 5 out of 10 questions right is approximately 0.2461, or 24.61%.