A student randomly guesses on 10 true or false
questions. Use the binomial model to find the probability
that the student gets 7 out of the 10 questions right.
P (x) = [ ]
n!
x!(n−x)!
p
xq
n−x
11.8%
20.9%
32.4%
50%
The probability that the student gets 7 out of the 10 questions right using the binomial model is 32.4%.
This is calculated using the formula for the binomial probability distribution:
P(x) = (nCx) * p^x * q^(n-x)
where:
- P(x) is the probability of getting exactly x successes
- n is the total number of trials (in this case, 10 questions)
- x is the number of successes (in this case, 7 questions right)
- p is the probability of success on each trial (in this case, 0.5 because the student randomly guesses)
- q is the probability of failure on each trial (in this case, also 0.5 because it is the complement of p, since there are only true/false options)
Using the formula, we can plug in the values:
P(7) = (10C7) * 0.5^7 * 0.5^(10-7)
= (10!)/(7!(10-7)!) * 0.5^7 * 0.5^3
= (10!)/(7!3!) * (0.5^7 * 0.5^3)
= (10*9*8)/(3*2*1) * (0.5^7 * 0.5^3)
= 120 * (0.5^7 * 0.5^3)
= 120 * 0.5^10
= 120 * 0.0009765625
= 0.1171875
≈ 11.8%
Therefore, the probability that the student gets 7 out of the 10 questions right is approximately 11.8%.