Question

7. Sky Castle Company is conducting research on 3 independent communities of 12 members each to see how they have been consuming calories for the past few years. Sky Castle has conducted an experiment to categorize community members to be either high risk participants or low risk participants and that each member in each community has a 30% chance to be categorized as high risk participants. What is the probability that at least 11 members are categorized as low risk participants in two of three communities?

Answer 1

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.

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

In which $C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

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 $n = 12$

30% chance to be categorized as high risk participants. So 100-30 = 70% probability of being categorized as low risk participants. So $p = 0.7$

This probability is

$P(X \leq 11) = P(X = 11) + P(X = 12)$

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 1) = C_{12,11}.(0.7)^{11}.(0.3)^{1} = 0.0712$

$P(X = 2) = C_{12,12}.(0.7)^{12}.(0.3)^{0} = 0.0138$

$P(X \leq 11) = P(X = 11) + P(X = 12) = 0.0712 + 0.0138 = 0.0850$

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 $p = 0.085$

Three comunities, so $n = 3$

This probability is P(X = 2).

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 2) = C_{3,2}.(0.085)^{2}.(0.915)^{1} = 0.0198$

1.98% probability that at least 11 members are categorized as low risk participants in two of three communities