A mail order company has 8% success rate. If it mails advertisements to 600 people, find the probability of getting less than 40 sales.
Find mean and standard deviation.
mean = np = 600 * .08 = ?
standard deviation = √npq = √(600)(.08)(.92) = ?
Note: q = 1 - p
I'll let you finish the calculations.
Once you have the mean and standard deviation, then use z-scores:
z = (x - mean)/sd
x = 40
Once you calculate the z-score, use the z-table to find your probability. (Remember that the question is asking for the probability of "getting less than 40" sales.)
I hope this will help get you started.
.1075
To find the probability of getting less than 40 sales, we need to use the binomial distribution.
The formula for the binomial distribution is:
P(x) = C(n, x) * p^x * (1 - p)^(n - x)
where:
P(x) is the probability of getting exactly x successes
n is the number of trials (in this case, 600 people)
x is the number of successful outcomes (sales)
p is the probability of success (8% or 0.08)
C(n, x) represents the binomial coefficient, which is calculated as n! / (x! * (n - x)!)
We want to find the probability of getting less than 40 sales, so we need to calculate the probabilities for all values from 0 to 39 and sum them up.
P(less than 40 sales) = P(0 sales) + P(1 sale) + P(2 sales) + ... + P(39 sales)
Let's calculate this step-by-step:
1. Calculate P(0 sales):
P(0) = C(600, 0) * 0.08^0 * (1 - 0.08)^(600 - 0)
To calculate C(600, 0), we have 0! in the denominator, which is equal to 1. So, C(600, 0) = 1.
P(0) = 1 * 1 * (0.92)^600 ≈ 0.012
2. Calculate P(1 sale):
P(1) = C(600, 1) * 0.08^1 * (1 - 0.08)^(600 - 1)
C(600, 1) = 600! / (1! * (600 - 1)!), which simplifies to 600.
P(1) = 600 * 0.08 * (0.92)^599 ≈ 0.052
3. Calculate P(2 sales):
P(2) = C(600, 2) * 0.08^2 * (1 - 0.08)^(600 - 2)
C(600, 2) = 600! / (2! * (600 - 2)!), which simplifies to 178,200.
P(2) = 178,200 * 0.08^2 * (0.92)^598 ≈ 0.127
Continue this process for all values from 0 to 39 and sum up the probabilities to get the final result.
To find the probability of getting less than 40 sales, we need to use the binomial probability formula. The binomial probability formula is given by:
P(X = k) = (n C 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, p is the probability of success in a single trial, (n C k) represents the binomial coefficient (n choose k), and ^(n - k) represents the exponentiation of (1 - p) to the power of (n - k).
In this case, the number of trials (n) is 600, the probability of success (p) is 0.08 (8% as a decimal), and we want to find the probability of getting less than 40 successes.
To find the probability of getting less than 40 sales, we need to sum up the probabilities of getting 0, 1, 2, 3, ..., 39 sales.
The probability of getting exactly k sales is given by the binomial probability formula. We will use a combination of a calculator or a spreadsheet software like Microsoft Excel to calculate the binomial probabilities for each value of k from 0 to 39 and then sum them up.
In Excel, you can use the BINOM.DIST function to calculate the binomial probabilities. The formula to calculate the probability of getting less than 40 sales would look like this:
=P(X=0) + P(X=1) + P(X=2) + ... + P(X=39)
Using this formula, you can input the values of n (600), p (0.08), and the range of values for k (0 to 39) to get the probability of getting less than 40 sales.
The result will be the probability of getting less than 40 sales as a decimal or a percentage.