Suppose that 20% of the population uses Brand X laundry soap. If 9 customers at a supermarket are surveyed, what is the probability that 3 of them use Brand X?
To calculate the probability that 3 out of 9 surveyed customers use Brand X laundry soap, you need to use the binomial probability formula. The binomial probability formula is:
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
n is the number of trials or observations
In this case, n = 9 (since 9 customers are surveyed), x = 3 (since we want exactly 3 customers to use Brand X), and p = 0.20 (since the probability of using Brand X is 20%).
Now, let's calculate the probability:
P(3) = C(9, 3) * (0.20)^3 * (1 - 0.20)^(9 - 3)
C(9, 3) = 9! / (3! * (9-3)!) (C(n, x) formula)
C(9, 3) = (9 * 8 * 7) / (3 * 2 * 1) = 84
P(3) = 84 * (0.20)^3 * (0.80)^6
P(3) = 0.2059 or 20.59%
Therefore, the probability that exactly 3 out of 9 surveyed customers use Brand X laundry soap is approximately 20.59%.
To calculate the probability of getting exactly 3 customers out of 9 using Brand X laundry soap, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
where:
- P(X = k) is the probability of getting exactly k successes.
- n is the total number of trials.
- k is the number of successes.
- p is the probability of success in a single trial.
- (nCk) is the notation for the binomial coefficient, which represents the number of ways to choose k successes out of n trials.
In this case, n = 9 (total number of customers surveyed), k = 3 (number of customers using Brand X), and p = 0.20 (probability of using Brand X).
Let's calculate the probability using the formula:
P(X = 3) = (9C3) * 0.20^3 * (1-0.20)^(9-3)
To calculate (9C3), we can use the formula:
nCk = n! / (k! * (n-k)!)
(9C3) = 9! / (3! * (9-3)!)
Simplifying the expression:
9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
3! = 3 * 2 * 1
(9-3)! = 6 * 5 * 4 * 3 * 2 * 1
Substituting the values into the formula:
(9C3) = (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (6 * 5 * 4 * 3 * 2 * 1))
Calculating (9C3) value:
(9C3) = 84
Substituting the calculated values into the binomial probability formula:
P(X = 3) = 84 * 0.20^3 * (1-0.20)^(9-3)
Calculating the probability:
P(X = 3) = 84 * 0.20^3 * 0.80^6
P(X = 3) ≈ 0.261