In a population where
85% prefer Candidate A, an organization conducts a poll of 19 voters. Find the probability that 17 of the 19 voters will prefer Candidate A.
To find the probability that 17 of the 19 voters will prefer Candidate A, we will use the binomial probability formula.
The binomial probability formula is:
P(x) = (nCx) * p^x * q^(n-x)
where:
P(x) is the probability of x occurrences,
n is the total number of trials,
x is the number of successful outcomes,
p is the probability of a successful outcome,
q is the probability of a failure (1 - p), and
(nCx) is the combination formula, given by n! / (x!(n-x)!)
In this case:
n = 19 (total number of voters)
x = 17 (number of voters preferring Candidate A)
p = 0.85 (probability of a voter preferring Candidate A)
q = 1 - p = 1 - 0.85 = 0.15
Substituting the values into the formula:
P(17) = (19C17) * 0.85^17 * 0.15^(19-17)
To simplify the calculations, we can use a combination calculator to find (19C17). In this case, (19C17) = 19! / (17!(19-17)!) = 19! / (17! * 2!) = (19 * 18) / 2 = 171.
Substituting this value back into the formula:
P(17) = 171 * 0.85^17 * 0.15^2
Calculating the value:
P(17) ≈ 171 * 0.036709 * 0.0225
P(17) ≈ 1.012
Therefore, the probability that 17 of the 19 voters will prefer Candidate A is approximately 1.012 (or approximately 1.2%).
To find the probability that 17 out of 19 voters will prefer Candidate A, you can use the binomial probability formula. The formula is:
P(k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(k) represents the probability of getting exactly k successes,
- C(n, k) is the number of combinations of n items taken k at a time (also known as the binomial coefficient), and
- p is the probability of success in a single trial.
In this case, we have:
- n = 19 (the total number of voters)
- k = 17 (the number of voters who prefer Candidate A)
- p = 0.85 (the probability of preferring Candidate A)
First, let's calculate C(n, k):
C(n, k) = n! / (k! * (n - k)!)
C(19, 17) = 19! / (17! * (19 - 17)!)
= 19! / (17! * 2!)
= (19 * 18) / 2
= 19 * 9
= 171
Next, let's calculate p^k * (1 - p)^(n - k):
p^k * (1 - p)^(n - k) = 0.85^17 * (1 - 0.85)^(19 - 17)
= 0.85^17 * 0.15^2
Now, substitute the values into the binomial probability formula:
P(17) = C(19, 17) * p^k * (1 - p)^(n - k)
= 171 * (0.85^17 * 0.15^2)
Calculating this expression will give you the probability that 17 out of 19 voters will prefer Candidate A.
another binomial
p (prefer A) = .85 = (P)
p (not prefer A) = .15 = (N)
(P + N)^19 = 19C17 P^17 N^2
... = 171 * .85^17 * .15^2