A brand name has a 40% recognition rate. If the owner of the brand wants to verify that rate by beginning with a small sample of 10 randomly selected consumers, find the probability that exactly 4 of the 10 consumers recognize the brand name. Also find the probability that the number who recognizes the brand name is not 4.

I really need the correct formula for this problem.

To solve this problem, we can use the binomial probability formula.

The binomial probability formula is given by:
P(x) = (nCx) * p^x * (1-p)^(n-x).

Where:
P(x) is the probability of getting exactly x successes,
n is the total number of trials (in this case, the number of consumers),
x is the number of successes (in this case, the number of consumers who recognize the brand name),
p is the probability of success in any one trial (in this case, the recognition rate of the brand name), and
(1-p) is the probability of failure in any one trial.

Now let's calculate the probability that exactly 4 out of 10 consumers recognize the brand name:

n = 10 (the total number of consumers)
x = 4 (the number of consumers who recognize the brand name)
p = 0.4 (the recognition rate of the brand name)

P(4) = (10C4) * 0.4^4 * (1-0.4)^(10-4)

To calculate (10C4), we use the combination formula:
(10C4) = 10! / (4!(10-4)!)

Substituting the values into the formula, we get:

P(4) = (10! / (4!(10-4)!)) * 0.4^4 * (1-0.4)^(10-4)

Simplifying, we have:

P(4) = (10! / (4!6!)) * 0.4^4 * 0.6^6

P(4) = (10 * 9 * 8 * 7)/(4 * 3 * 2 * 1) * 0.4^4 * 0.6^6

P(4) = 210 * 0.4^4 * 0.6^6

Using a calculator, we can compute:

P(4) ≈ 0.2508

Therefore, the probability that exactly 4 out of 10 consumers recognize the brand name is approximately 0.2508.

To find the probability that the number who recognizes the brand name is not 4, we need to find the sum of probabilities of all other possibilities (0, 1, 2, 3, 5, 6, 7, 8, 9, 10) and subtract it from 1 (total probability).

P(not 4) = 1 - [P(0) + P(1) + P(2) + P(3) + P(5) + P(6) + P(7) + P(8) + P(9) + P(10)]

You can use the same formula mentioned earlier to calculate each individual probability and then sum them up to find P(not 4).

I hope this explanation helps you understand the problem and the formula.

To solve this problem, we can use the binomial probability formula.

The probability of exactly x successes in n trials, where the probability of success in each trial is p, is given by the formula:

P(x) = (nCx) * p^x * (1-p)^(n-x),

where nCx is the binomial coefficient, calculated as:

nCx = n! / (x! * (n-x)!),

and "!" denotes the factorial of a number.

In this case, n = 10 (the number of consumers selected), p = 0.40 (the recognition rate of the brand name), and we want to find the probability of exactly 4 consumers recognizing the brand name.

Using the formula:

P(4) = (10C4) * (0.40)^4 * (1-0.40)^(10-4)

P(4) = (10! / (4! * (10-4)!)) * (0.40)^4 * (0.60)^6

Calculating the terms:

P(4) = (10! / (4! * 6!)) * (0.40)^4 * (0.60)^6

P(4) = (10 * 9 * 8 * 7 * 6! / (4! * 6!)) * (0.40)^4 * (0.60)^6

P(4) = (10 * 9 * 8 * 7 / (4!)) * (0.40)^4 * (0.60)^6

Now, we can calculate the value of P(4) using a calculator or software. It comes out to be approximately 0.2508.

Therefore, the probability that exactly 4 of the 10 consumers recognize the brand name is 0.2508.

To find the probability that the number of consumers who recognize the brand name is not 4, we can subtract the probability of exactly 4 from 1 (since the sum of all probabilities equals 1):

P(not 4) = 1 - P(4)

P(not 4) = 1 - 0.2508

P(not 4) = 0.7492

Therefore, the probability that the number of consumers who recognize the brand name is not 4 is approximately 0.7492.

P(X=4)

p = .6; n=10) = 10C4 *.4^4 * .6^6
= 0.2508

P(x = not 4) = 1- 0.2508 = 0.7492