Anita's, a fast-food chain specializing in hot dogs and garlic fries, keeps track of the proportion of its customers who decide to eat in the restaurant (as opposed to ordering the food "to go"), so it can make decisions regarding the possible construction of in-store play areas, the attendance of its mascot Sammy at the franchise locations, and so on. Anita's reports that 52% of its customers order their food to go. If this proportion is correct, what is the probability that, in a random sample of 5 customers at Anita's, exactly 2 order their food to go?
2 is the required outcome,and the possible outcome=(0.52*5), Hence ; P(O)= 2/(0.52*5)=0.77
To solve this problem, we can use the binomial probability formula. The binomial probability formula calculates the probability of obtaining a specific number of successes in a fixed number of independent trials.
The formula is given by:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes,
- n is the number of trials (sample size),
- k is the number of successes (exactly 2 in this case),
- p is the probability of success in a single trial (52% or 0.52), and
- (n choose k) is the binomial coefficient, calculated as (n! / (k! * (n - k)!)).
Let's calculate the probability:
P(X = 2) = (5 choose 2) * (0.52^2) * (0.48^(5 - 2))
Calculating (5 choose 2):
(5! / (2! * (5 - 2)!)) = (5! / (2! * 3!)) = (5 * 4) / (2 * 1) = 10
Substituting the values back into the formula:
P(X = 2) = 10 * (0.52^2) * (0.48^3)
Now, we can calculate the probability:
P(X = 2) = 10 * 0.2704 * 0.110592
P(X = 2) ≈ 0.2995
Therefore, the probability that exactly 2 customers out of a random sample of 5 order their food to go is approximately 0.2995 or 29.95%.
To determine the probability of exactly 2 customers ordering their food to go out of a random sample of 5 customers, we can use the binomial probability formula.
The formula for calculating the binomial probability is:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where:
P(X=k) is the probability of getting exactly k successes,
n is the number of trials,
k is the number of desired successes,
p is the probability of success on each trial.
In this case, the probability of ordering the food to go (success) is p = 0.52, and we want exactly 2 customers out of 5 to order their food to go (k = 2).
Then, we can substitute these values into the formula:
P(X=2) = (5 choose 2) * (0.52)^2 * (1-0.52)^(5-2)
Calculating the values:
(5 choose 2) = 5! / (2! * (5-2)!) = 10
(0.52)^2 = 0.2704
(1-0.52)^(5-2) = 0.2197
Now, plug in the values and calculate:
P(X=2) = 10 * 0.2704 * 0.2197
P(X=2) ≈ 0.5946
Therefore, the probability of exactly 2 customers ordering their food to go out of a random sample of 5 customers is approximately 0.5946.