in a recent poll of city voters it was found that 90% liked the view of the city skyline.find the probability that exactly 2 voters in a sample of 5 voters will like the viewof the city skyline.
prob of liking = .9
prob of not liking = .1
prob that 2 of 5 will like = C(5,2)(.9)^2 (.1)^3)
= 10(.81)(.001) = .0081
To find the probability that exactly 2 voters in a sample of 5 voters will like the view of the city skyline, we can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (nCx) * p^x * (1-p)^(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 outcomes,
- p is the probability of success for a single trial, and
- (nCx) is the binomial coefficient calculated as: n! / (x! * (n-x)!) where ! denotes factorial.
In this case, we have:
- n = 5 (sample size),
- x = 2 (number of voters who like the view),
- p = 0.9 (probability of a voter liking the view).
Let's calculate the probability:
P(x = 2) = (5C2) * (0.9)^2 * (1-0.9)^(5-2)
To calculate the binomial coefficient (5C2), we use the formula:
(5C2) = 5! / (2! * (5-2)!)
Simplifying,
(5C2) = 5! / (2! * 3!)
Calculating the factorials:
5! = 5 * 4 * 3!
2! = 2 * 1
Substituting the values:
(5C2) = (5 * 4 * 3!) / (2! * 3!)
Canceling out the factorial terms:
(5C2) = (5 * 4) / 2
(5C2) = 10
Further calculations:
P(x = 2) = 10 * (0.9)^2 * (1-0.9)^(5-2)
P(x = 2) = 10 * (0.9)^2 * (0.1)^3
P(x = 2) = 10 * 0.81 * 0.001
P(x = 2) = 0.0081
So, the probability that exactly 2 voters in a sample of 5 voters will like the view of the city skyline is 0.0081, or 0.81%.