find the indicated probability.\n5) the brand name of a certain chain of coffee shops has a 53% recognition…
find the indicated probability.\n5) the brand name of a certain chain of coffee shops has a 53% recognition rate in the town of coffleton. an executive from the company wants to verify the recognition rate as the company is interested in opening a coffee shop in the town. he selects a random sample of 10 coffleton residents. find the probability that the number that recognize the brand name is not 4.\na) 0.000851\nb) 0.0905\nc) 0.821\nd) 0.179
Answer
Explanation:
Step1: Identify binomial parameters
This is a binomial - probability problem. The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single - trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 10$, $p=0.53$, and $1 - p = 0.47$.
Step2: Calculate $P(X = 4)$
First, calculate the combination $C(10,4)=\frac{10!}{4!(10 - 4)!}=\frac{10!}{4!6!}=\frac{10\times9\times8\times7}{4\times3\times2\times1}=210$. Then, $P(X = 4)=C(10,4)\times(0.53)^{4}\times(0.47)^{6}=210\times0.07890481\times0.01528477=210\times0.00119677 = 0.179$.
Step3: Calculate $P(X\neq4)$
We know that $P(X\neq4)=1 - P(X = 4)$. Since $P(X = 4)=0.179$, then $P(X\neq4)=1 - 0.179 = 0.821$.
Answer:
C. 0.821