According to a previous report, 15% of households had some type of high-speed Internet connection. Suppose 12 households are selected at random and the number of households with high-speed Internet is recorded. (a) Find the probability that exactly 6 households have high-speed Internet.
binomial
p = .15
1-p = .85
C(12,6) = 924
so
924 * .15^6 * .85^6
To find the probability of exactly 6 households having high-speed Internet out of the 12 selected households, we need to use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (nCk) * p^k * q^(n-k)
Where:
- P(X = k) is the probability of exactly k successes
- n is the total number of trials (in this case, the number of households selected)
- k is the number of successful outcomes (in this case, the number of households with high-speed Internet)
- p is the probability of a single trial being a success (in this case, the probability that a household has high-speed Internet)
- q is the probability of a single trial being a failure (in this case, the probability that a household does not have high-speed Internet)
- (nCk) represents the combination formula, which calculates the number of ways to choose k items out of n without regard to order.
In this scenario, the probability of a household having high-speed Internet is given as 15%, which can be expressed as 0.15. Therefore, p = 0.15.
The probability of a household not having high-speed Internet can be calculated as 1 - p = 1 - 0.15 = 0.85. Therefore, q = 0.85.
Now we can substitute the values into the formula:
P(X = 6) = (12C6) * (0.15)^6 * (0.85)^(12-6)
To calculate (12C6) or the combination formula, you can use the formula:
(12C6) = 12! / (6!(12-6)!)
Where "!" denotes the factorial function.
After calculating the combination formula and evaluating the rest of the expression, you will find the probability of exactly 6 households having high-speed Internet.