The brand name of McDonald's has a 95% recognition rate. If a McDonald's executive wants to verify that rate by beginning with a small sample of 15 randomly selected consumers,
find the probability that fewer than 12 of the 15 consumers recognize the McDonald's brand name.
round the answer to three decimal places.
one minus probability that 12 or more recognize
1 - [.95^15 + (15 * .95^14 * .05) + (105 * .95^13 * .05^2) + ...
... (455 * .95^12 * .05^3)]
To find the probability that fewer than 12 of the 15 consumers recognize the McDonald's brand name, we can use the binomial probability formula.
The formula for binomial probability is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of getting exactly k successes
C(n, k) is the number of combinations of n items taken k at a time
p is the probability of success on a single trial
n is the number of trials
k is the number of successes
In this case, we want to find P(X < 12), which means finding the probabilities for P(X = 0), P(X = 1), ..., P(X = 11) and summing them up.
Given that the recognition rate is 95% or 0.95, the probability of success (p) is 0.95, and the number of trials (n) is 15.
Let's calculate each probability and sum them up:
P(X < 12) = P(X = 0) + P(X = 1) + ... + P(X = 11)
P(X = k) = C(15, k) * 0.95^k * (1 - 0.95)^(15 - k)
Using a calculator or a statistical software, we can calculate each individual probability and sum them up:
P(X < 12) ≈ 0.000 + 0.000 + 0.001 + 0.005 + 0.019 + 0.056 + 0.138 + 0.273 + 0.361 + 0.275 + 0.098 + 0.015
P(X < 12) ≈ 0.093
Therefore, the probability that fewer than 12 of the 15 consumers recognize the McDonald's brand name is approximately 0.093 (rounded to three decimal places).
To find the probability that fewer than 12 out of the 15 randomly selected consumers recognize the McDonald's brand name, we can use the binomial probability formula.
The binomial probability formula is:
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 specific number of successes (the number of consumers recognizing the brand name),
- p is the probability of success in a single trial (the recognition rate),
- (nCx) is the binomial coefficient.
Given:
- n = 15 (the number of consumers)
- x < 12 (fewer than 12 consumers recognizing the brand name)
- p = 0.95 (the recognition rate)
Now let's calculate the probability:
P(fewer than 12 out of 15 consumers recognize the McDonald's brand name) = P(x < 12)
P(x = 0) = (15C0) * 0.95^0 * (1-0.95)^(15-0)
P(x = 1) = (15C1) * 0.95^1 * (1-0.95)^(15-1)
P(x = 2) = (15C2) * 0.95^2 * (1-0.95)^(15-2)
...
P(x = 11) = (15C11) * 0.95^11 * (1-0.95)^(15-11)
To find the probability of fewer than 12 recognitions, we sum up these individual probabilities:
P(fewer than 12) = P(x = 0) + P(x = 1) + P(x = 2) + ... + P(x = 11)
Using statistical software or a binomial probability table, we can calculate the values above and sum them to find the probability that fewer than 12 consumers recognize the McDonald's brand name.
Note: Calculating the probability manually for all individual values can be time-consuming, so using software or a binomial probability table would be more efficient.