a brand name has a 50% recognition rate assume the owner of the brand wants to verify that rate by beginning with a small sample of 5 randomly selected consumers
the probability that exactly 4 of the 5 consumers recognize the brand name
sdfs
To find the probability that exactly 4 out of the 5 randomly selected consumers recognize the brand name, we can use the binomial probability formula.
The formula to calculate the probability of x successes in n trials, where p is the probability of success, is:
P(x) = (nCx) * (p^x) * ((1-p)^(n-x))
In this case, we have:
n = 5 (the number of trials or consumers in the sample)
x = 4 (the number of successes or consumers who recognize the brand name)
p = 0.5 (the recognition rate or probability of success)
To find the binomial coefficient (nCx), we use the formula:
(nCx) = n! / (x! * (n-x)!)
In this case, we have:
n = 5 (the number of trials)
Calculating (nCx):
(5C4) = 5! / (4! * (5-4)!)
= 5! / (4! * 1!)
= 5
Putting all the values into the formula, we have:
P(4) = (5C4) * (0.5^4) * (0.5^(5-4))
= 5 * (0.5^4) * (0.5^1)
= 5 * 0.0625 * 0.5
= 0.15625
Therefore, the probability that exactly 4 out of the 5 randomly selected consumers recognize the brand name is 0.15625, or 15.625%.