39% of households have some sort of high speed internet service. 15 homes are selected at random. the probability that 5 homes will have high speed service is?
To find the probability that exactly 5 out of 15 randomly selected homes have high-speed internet service, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (nCk) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability that exactly k successes occur
n is the total number of trials (in this case, 15 homes selected)
k is the number of desired successes (in this case, 5 homes with high-speed service)
p is the probability of success (in this case, the percentage of households with high-speed service, which is 39% or 0.39)
nCk is the combination formula, also denoted as "n choose k", which calculates the number of ways to choose k items from a collection of n items. It is calculated as n! / (k! * (n - k)!)
Now, let's plug in the values into the formula:
P(X = 5) = (15C5) * (0.39)^5 * (1 - 0.39)^(15 - 5)
To calculate (15C5), we use the combination formula:
15C5 = 15! / (5! * (15 - 5)!) = 3003
Now, we can substitute the values back into the formula:
P(X = 5) = 3003 * (0.39)^5 * (0.61)^10
Using a calculator, we can evaluate this expression to find the final probability.