Answer:
1.98% probability that at least 11 members are categorized as low risk participants in two of three communities
Step-by-step explanation:
To solve this question, we need to use two separate binomial trials.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
In which
is the number of different combinations of x objects from a set of n elements, given by the following formula.
![C_{n,x} = \frac{n!}{x!(n-x)!}](https://tex.z-dn.net/?f=C_%7Bn%2Cx%7D%20%3D%20%5Cfrac%7Bn%21%7D%7Bx%21%28n-x%29%21%7D)
And p is the probability of X happening.
Probability of at least 11 members being categorized as low risk participants.
12 members in the sample, so ![n = 12](https://tex.z-dn.net/?f=n%20%3D%2012)
30% chance to be categorized as high risk participants. So 100-30 = 70% probability of being categorized as low risk participants. So ![p = 0.7](https://tex.z-dn.net/?f=p%20%3D%200.7)
This probability is
![P(X \leq 11) = P(X = 11) + P(X = 12)](https://tex.z-dn.net/?f=P%28X%20%5Cleq%2011%29%20%3D%20P%28X%20%3D%2011%29%20%2B%20P%28X%20%3D%2012%29)
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
![P(X = 1) = C_{12,11}.(0.7)^{11}.(0.3)^{1} = 0.0712](https://tex.z-dn.net/?f=P%28X%20%3D%201%29%20%3D%20C_%7B12%2C11%7D.%280.7%29%5E%7B11%7D.%280.3%29%5E%7B1%7D%20%3D%200.0712)
![P(X = 2) = C_{12,12}.(0.7)^{12}.(0.3)^{0} = 0.0138](https://tex.z-dn.net/?f=P%28X%20%3D%202%29%20%3D%20C_%7B12%2C12%7D.%280.7%29%5E%7B12%7D.%280.3%29%5E%7B0%7D%20%3D%200.0138)
![P(X \leq 11) = P(X = 11) + P(X = 12) = 0.0712 + 0.0138 = 0.0850](https://tex.z-dn.net/?f=P%28X%20%5Cleq%2011%29%20%3D%20P%28X%20%3D%2011%29%20%2B%20P%28X%20%3D%2012%29%20%3D%200.0712%20%2B%200.0138%20%3D%200.0850)
What is the probability that at least 11 members are categorized as low risk participants in two of three communities?
For each community, 8.50% probability of at least 11 members being categorized as low risk. So ![p = 0.085](https://tex.z-dn.net/?f=p%20%3D%200.085)
Three comunities, so ![n = 3](https://tex.z-dn.net/?f=n%20%3D%203)
This probability is P(X = 2).
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
![P(X = 2) = C_{3,2}.(0.085)^{2}.(0.915)^{1} = 0.0198](https://tex.z-dn.net/?f=P%28X%20%3D%202%29%20%3D%20C_%7B3%2C2%7D.%280.085%29%5E%7B2%7D.%280.915%29%5E%7B1%7D%20%3D%200.0198)
1.98% probability that at least 11 members are categorized as low risk participants in two of three communities