Forty-four percent of customers who visit a department store make a purchase. The probability, rounded to four decimal places, that in a random sample of 9 customers who will visit this department store, exactly 6 will make a purchase is:
P(x = 6) = nCx *p^x (1-p)^n-x
9C6(.44)^6(1-.44)^n-x
9C6(.44)(.56)^3
= 0.1070
To find the probability of exactly 6 customers making a purchase out of 9, we can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (nCx) * p^x * q^(n-x)
Where:
P(x) is the probability of exactly x successes
n is the number of trials
x is the number of successes
p is the probability of success
q is the probability of failure (1 - p)
nCx is the number of combinations of n objects taken x at a time
In this case, n = 9 (number of customers in sample),
x = 6 (number of customers making a purchase),
p = 0.44 (probability of making a purchase),
q = 1 - p = 1 - 0.44 = 0.56 (probability of not making a purchase).
Using the binomial probability formula, we can calculate:
P(6) = (9C6) * (0.44^6) * (0.56^3)
Calculating the combination (9C6):
(9C6) = 9! / (6! * (9-6)!)
= 9! / (6! * 3!)
= (9 * 8 * 7) / (3 * 2 * 1)
= 84
Substituting the values into the formula:
P(6) = (84) * (0.44^6) * (0.56^3)
Calculating this will give us the required probability.
To calculate the probability that exactly 6 out of 9 customers will make a purchase, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k))
Where:
- P(X = k) is the probability of exactly k successes
- n is the total number of trials
- p is the probability of success in a single trial
- (n choose k) is the binomial coefficient, also known as "n choose k"
Here, n = 9 (number of customers), p = 0.44 (probability of a customer making a purchase), and k = 6 (number of successes).
Using the formula, we can calculate the probability:
P(X = 6) = (9 choose 6) * (0.44^6) * ((1-0.44)^(9-6))
Now, let's calculate each part of the formula:
- (9 choose 6) can be calculated as: (9!)/(6!(9-6)!) = 84
- (0.44^6) = 0.0030458 (rounded to 4 decimal places)
- ((1-0.44)^(9-6)) = 0.1052694 (rounded to 4 decimal places)
Putting it all together:
P(X = 6) = 84 * 0.0030458 * 0.1052694
P(X = 6) = 0.2576 (rounded to 4 decimal places)
Therefore, the probability, rounded to four decimal places, that in a random sample of 9 customers who will visit this department store, exactly 6 will make a purchase is 0.2576.