A store manager believes that 42% of her customers are repeat business (have visited her store within the last two weeks). Assuming she is correct, what is the probability that out of the next 500 customers, between 200 and 250 customers are repeat business?
500 x .42 = ??
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To solve this problem, we will use the binomial probability formula. The formula is as follows:
P(x) = (nCx) * (p^x) * (q^(n-x))
where:
P(x) is the probability of getting exactly x successes
n is the number of trials
p is the probability of success
q is the probability of failure, which is equal to 1-p
nCx is the number of combinations of n items taken x at a time
In this case:
n = 500 (number of customers)
x ranges from 200 to 250 (number of repeat business customers)
p = 0.42 (probability of a customer being a repeat business)
q = 1 - p = 1 - 0.42 = 0.58 (probability of a customer not being a repeat business)
Now, let's calculate the probabilities for each value of x and sum them up to find the probability that between 200 and 250 customers are repeat business:
P(200) = (500C200) * (0.42^200) * (0.58^300)
P(201) = (500C201) * (0.42^201) * (0.58^299)
...
P(250) = (500C250) * (0.42^250) * (0.58^250)
Let's calculate them step by step:
Step 1: Calculate nCx value:
nCx = n! / (x! * (n - x)!)
nCx = 500! / (200! * (500 - 200)!)
Using a calculator or computer program, this value can be calculated as:
nC200 = 2.24891059e+119
Step 2: Calculate each probability value using the formula:
P(200) = 2.24891059e+119 * (0.42^200) * (0.58^300)
P(201) = 2.24891059e+119 * (0.42^201) * (0.58^299)
...
P(250) = 2.24891059e+119 * (0.42^250) * (0.58^250)
Let's evaluate P(200) step by step:
P(200) = 2.24891059e+119 * (0.42^200) * (0.58^300)
P(200) = 2.24891059e+119 * (0.42^200) * (1.86922016e-82)
Using a calculator or computer program, this value can be calculated as:
P(200) = 7.02131578e-40
Similarly, we can calculate the other probability values for x ranging from 201 to 250.
Lastly, we sum up all the probabilities from P(200) to P(250) to get the final probability:
P(final) = P(200) + P(201) + ... + P(250)
Obtaining this sum would require adding the calculated probabilities together.
To find the probability that between 200 and 250 customers out of the next 500 are repeat business, we can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (nCk) * p^k * (1 - p)^(n - k)
Where:
- P(x) is the probability of having x successes
- n is the number of trials (500 in this case)
- k is the number of successes (between 200 and 250 in this case)
- p is the probability of success (42% or 0.42 in this case)
- (nCk) is the number of combinations, which can be calculated using the formula:
nCk = n! / (k! * (n - k)!)
Now let's calculate the probability for each value of k between 200 and 250 and sum them up.
P_total = P(X = 200) + P(X = 201) + ... + P(X = 250)
Where P(X = k) is the probability of having k repeat business customers.
To calculate the probabilities, we need to substitute the values into the formula. However, calculating individual probabilities for a range of values can be time-consuming. Instead, we can use a statistical software or a calculator that provides binomial probability calculations like Excel, Python, or an online calculator.
For example, using Excel, you can use the formula:
=SUMPRODUCT(BINOM.DIST.RANGE(200, 250, 0.42, 1))
This formula calculates the sum of probabilities between 200 and 250 given a success probability of 0.42.
By substituting the values into the formula and performing the calculations, you can find the probability that between 200 and 250 customers out of the next 500 are repeat business.