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.
P(x) = [n!/x!(n_x)!]P^xQ^n-x
To determine the probability that the student gets 5 out of 10 questions right, we can use the binomial probability formula:
P(x) = [n!/x!(n-x)!] * P^x * Q^(n-x)
where:
- P(x) represents the probability of getting exactly x successes (in this case, getting 5 questions right)
- n is the total number of trials (in this case, 10 questions)
- x is the number of successes (in this case, getting 5 questions right)
- P is the probability of success on a single trial (in this case, getting a question right, which is 1/2 as it is a true/false question)
- Q is the probability of failure on a single trial (in this case, getting a question wrong, which is also 1/2)
Now let's calculate the probability:
P(x) = [10!/5!(10-5)!] * (1/2)^5 * (1/2)^(10-5)
P(x) = [10!/5!5!] * (1/2)^5 * (1/2)^5
P(x) = [10 * 9 * 8 * 7 * 6 / (5 * 4 * 3 * 2 * 1)] * (1/32) * (1/32)
P(x) = [30240 / 120] * (1/1024)
P(x) = 252 * (1/1024)
P(x) ≈ 0.24609375
Therefore, the probability that the student gets exactly 5 out of 10 questions right is approximately 0.24609375 or 24.61%.