A student guesses on 10 true or false questions. Use the binomial model to find the probability that the student gets 7 out of 10 questions right.
P(x)=[n!/x!(n-x)!]p^xq^n-x
In this case, we will use the binomial distribution formula:
P(x) = [n! / (x!(n-x)!)] * p^x * q^(n-x)
Where:
P(x) is the probability of getting exactly x successes
n is the total number of trials
x is the number of successful trials
p is the probability of success on a single trial
q is the probability of failure on a single trial (q = 1-p)
In this case, the student has a 1/2 chance of getting each question right (p=1/2) and a 1/2 chance of getting each question wrong (q=1/2).
n = 10 (total number of questions)
x = 7 (number of correct answers)
p = 1/2 (probability of getting a question right)
q = 1/2 (probability of getting a question wrong)
Plugging in these values into the binomial distribution formula:
P(7) = [10! / (7!(10-7)!)] * (1/2)^7 * (1/2)^(10-7)
= [10! / (7!3!)] * (1/2)^7 * (1/2)^3
= [10 * 9 * 8 / (3 * 2 * 1)] * (1/2)^7 * (1/2)^3
= 120 * (1/2)^10
= 120 * 1/1024
= 0.1171875
Therefore, the probability that the student gets exactly 7 out of 10 questions right is approximately 0.1172, or 11.72%.