80% of the American population likes pizza. If 8 Americans are chosen at random, find the probability that 6 of them like pizza.
n=8, p=.80, q=1-.80=0.20, x=6
P(x)=nCxPxQn-x
P(x)=8C6(0.80)6(0.20)2
= (28)x(0.262144)x(0.04)
=0.294
To find the probability that 6 out of 8 randomly chosen Americans like pizza, we can use the binomial probability formula. The probability of success (liking pizza) is given as 80% or 0.8, and the probability of failure (not liking pizza) is 1 - probability of success = 0.2.
The binomial probability formula is:
P(X=k) = C(n,k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of getting k successes
- C(n,k) is the number of combinations of n items taken k at a time
- p is the probability of success
- n is the total number of trials
In this case, n = 8, k = 6, p = 0.8, and (1-p) = 0.2. So, let's calculate the probability:
P(X=6) = C(8,6) * 0.8^6 * 0.2^(8-6)
Now, let's calculate each part separately:
C(8,6) = 8! / (6! * (8-6)!) = 8! / (6! * 2!) = (8 * 7) / (2 * 1) = 28
0.8^6 = 0.262144
0.2^(8-6) = 0.2^2 = 0.04
Now, let's substitute the values into the formula:
P(X=6) = 28 * 0.262144 * 0.04
P(X=6) ≈ 0.2936
Therefore, the probability that 6 out of 8 randomly chosen Americans like pizza is approximately 0.2936 or 29.36%.
To find the probability that 6 out of 8 randomly chosen Americans like pizza, we can use the binomial probability formula. The formula is:
P(X = k) = (nCk) * (p^k) * (q^(n-k))
Where:
- P(X = k) is the probability of getting exactly k successes (in this case, picking Americans who like pizza),
- n is the total number of trials (in this case, the number of Americans chosen),
- k is the number of successes (in this case, 6 Americans liking pizza),
- p is the probability of success in a single trial (in this case, the probability of an American liking pizza),
- (nCk) is the binomial coefficient, which represents the number of ways to choose k successes out of n trials, and
- q is the probability of failure in a single trial (in this case, the probability of an American not liking pizza).
Given that 80% of the American population likes pizza, we can say that the probability of an American liking pizza (p) is 0.80. Therefore, the probability of an American not liking pizza (q) is 1 - p = 1 - 0.80 = 0.20.
Now, let's substitute these values into the formula:
P(X = 6) = (8C6) * (0.80^6) * (0.20^(8-6))
Calculating the values:
(8C6) = 8! / (6! * (8-6)!) = 28
(0.80^6) = 0.262144
(0.20^2) = 0.04
P(X = 6) = 28 * 0.262144 * 0.04
P(X = 6) ≈ 0.2949
Therefore, the probability that exactly 6 out of 8 randomly chosen Americans like pizza is approximately 0.2949 or 29.49%.