Binomial problems: Basic
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 48% of its customers order their food to go. If this proportion is correct, what
is the probability that, in a random sample of 4 customers at Anita's, exactly 3 order their
food to go?
0.37
This is a binomial distribution:
p = .48, q = 1 - p = .52
The probability P{X} of exactly 4 customers from 5, will order food to go is:
P{X = 4} = (5 C 4)((.48)^4)((1 - .48)^1)
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 45% of its customers order their food to go. Suppose that this proportion is correct and that a random sample of 40 individual customers is taken.
Estimate the number of customers in the sample who order their food to go by giving the mean of the relevant distribution (that is, the expectation of the relevant random variable). Do not round your response.
Quantify the uncertainty of your estimate by giving the standard deviation of the distribution. Round your response to at least three decimal places.
To find the probability that exactly 3 out of 4 customers order their food to go, we can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = C(n, x) * p^x * (1 - p)^(n - x)
where:
P(x) is the probability of getting exactly x successes,
C(n, x) is the number of combinations of n items taken x at a time,
p is the probability of success on a single trial, and
n is the total number of trials.
In this case, we have:
n = 4 (the total number of customers in the sample)
x = 3 (the number of customers who order their food to go)
p = 0.48 (the proportion of customers who order their food to go)
First, let's calculate the number of combinations of 4 customers taken 3 at a time:
C(4, 3) = 4! / (3!(4-3)!) = 4
Next, substitute these values into the binomial probability formula:
P(3) = C(4, 3) * (0.48)^3 * (1 - 0.48)^(4 - 3)
= 4 * (0.48)^3 * (1 - 0.48)^(4 - 3)
= 4 * 0.48^3 * 0.52^1
Now, plug this into a calculator to find the probability:
P(3) ≈ 0.1814
So, the probability that exactly 3 out of 4 customers at Anita's order their food to go is approximately 0.1814 or 18.14%.