The probability that a student uses the Academic Resource Center on a regular basis is 0.26. In a group of 18 students, what is the probability that exactly 8 of them use the Academic Resource Center on a regular basis? Round your answer to 2 decimal places.
To calculate the probability that exactly 8 of the 18 students use the Academic Resource Center on a regular basis, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (n C k) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of exactly k successes
n is the number of trials or students in this case (18)
k is the number of successes or students using the Academic Resource Center (8)
p is the probability of success or the probability that a student uses the Academic Resource Center on a regular basis (0.26)
(1-p) is the probability of failure or the probability that a student does not use the Academic Resource Center on a regular basis (1 - 0.26 = 0.74)
(n C k) is the number of combinations of n items taken k at a time, calculated as n! / (k! * (n-k)!)
Plugging in the values, we get:
P(X = 8) = (18 C 8) * (0.26)^8 * (0.74)^(18-8)
Calculating this expression, we get:
P(X = 8) = (18! / (8! * (18-8)!) * (0.26)^8 * (0.74)^10
Simplifying this expression, we get:
P(X = 8) = 43758 * (0.26)^8 * (0.74)^10
Calculating this expression further, we get:
P(X = 8) ≈ 0.0529 (rounded to 2 decimal places)
Therefore, the probability that exactly 8 of the 18 students use the Academic Resource Center on a regular basis is approximately 0.05.
To find the probability that exactly 8 out of 18 students use the Academic Resource Center on a regular basis, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) represents the probability of getting exactly k successes (in this case, exactly 8 students using the Academic Resource Center on a regular basis).
n is the total number of trials (in this case, the total number of students, which is 18).
k is the number of successes we are interested in (in this case, 8 students using the Academic Resource Center on a regular basis).
p is the probability of success on each individual trial (in this case, the probability that a student uses the Academic Resource Center on a regular basis, which is 0.26).
C(n, k) is the binomial coefficient and represents the number of ways to choose k successes from n trials.
Let's calculate the probability using this formula:
P(X = 8) = C(18, 8) * (0.26)^8 * (1 - 0.26)^(18 - 8)
C(18, 8) = 18! / (8! * (18 - 8)!)
= 43758 / (40320 * 10)
= 0.135
Now, substitute the values into the formula:
P(X = 8) = 0.135 * (0.26)^8 * (1 - 0.26)^(18 - 8)
Calculating this expression will give us the probability of exactly 8 out of 18 students using the Academic Resource Center on a regular basis. Round the answer to 2 decimal places.
Find P(8).
n = 18
x = 8
p = 0.26
Use a binomial probability table or a calculator.
If you must do this by hand, here is a formula:
P(x) = (nCx)(p^x)[q^(n-x)]
I'll let you take it from here.