An insurer offers a health plan to the employees of a large company. As part of this plan, the individual employees may choose two of the supplementary covaerges A, B and C or they may choose no supplementary coverage. The proportions of the company's employees that choose coverage A, B and C are 1/4 ,1/3 and 5/12 respectively. Determine the probability that a randomly chosen employee will choose no supplementary coverage.

Question

An insurer offers a health plan to the employees of a large company. As part of this plan, the individual employees may choose two of the supplementary covaerges A, B and C or they may choose no supplementary coverage. The proportions of the company's employees that choose coverage A, B and C are 1/4 ,1/3 and 5/12 respectively. Determine the probability that a randomly chosen employee will choose no supplementary coverage.

Answer 1

I've got a different answer from the Actuary exam guide. You need to make a Venn Diagram and make x,y, and z into the middle portions of the diagram between AB, AC, and BC. From the question you know that x+y=1/4, x+z=1/3 and y+z=5/12. Adding these equations gives

(x+y)+(x+z)+(y+z)=1/4+1/3+5/12
then, 2(x+y+z)=1/2
Since you're looking for 1-(x+y+z) then 1-1/2=1/2. The probability is 1/2

Answer 2

To determine the probability that a randomly chosen employee will choose no supplementary coverage, we need to calculate the proportion of employees that choose each supplementary coverage and subtract it from 1.

Given that the proportions of employees that choose coverage A, B, and C are 1/4, 1/3, and 5/12 respectively, we can calculate the proportion of employees that choose no supplementary coverage as follows:

Proportion choosing no coverage = 1 - Proportion choosing coverage A - Proportion choosing coverage B - Proportion choosing coverage C

Proportion choosing no coverage = 1 - (1/4) - (1/3) - (5/12)

Now, let's simplify this expression:

Proportion choosing no coverage = 1 - 3/12 - 4/12 - 5/12

Proportion choosing no coverage = 1 - (12/12)

Proportion choosing no coverage = 0/12

Therefore, the probability that a randomly chosen employee will choose no supplementary coverage is 0.

Answer 3

To determine the probability that a randomly chosen employee will choose no supplementary coverage, we need to find the proportion of employees that choose no coverage.

Let's assign variables to represent the proportions of employees choosing each supplementary coverage:
- Let p(A) be the proportion choosing coverage A, which is given as 1/4.
- Let p(B) be the proportion choosing coverage B, which is given as 1/3.
- Let p(C) be the proportion choosing coverage C, which is given as 5/12.

We can calculate the proportion of employees that choose no supplementary coverage by subtracting the sum of the proportions of those who choose coverage A, B, and C from 1 (since the probabilities must add up to 1).

Let p(no coverage) represent the proportion of employees choosing no supplementary coverage:
p(no coverage) = 1 - (p(A) + p(B) + p(C))

Substituting the given values, we have:
p(no coverage) = 1 - (1/4 + 1/3 + 5/12)

To simplify the calculation, we need to find a common denominator for the fractions:
p(no coverage) = 1 - (3/12 + 4/12 + 5/12)

Combine the fractions:
p(no coverage) = 1 - (12/12)

The fraction 12/12 represents the entire company, and subtracting it means none of the employees choose no coverage. Therefore, the probability is 0.

So, the probability that a randomly chosen employee will choose no supplementary coverage is 0.

Answer 4

Choices are AB, AC, BC, None

Pr(A and B)= Pr(A)Pr(B)= 1/4*1/3= 1/12
Pr(B and C)= Pr(B)Pr(C)= 1/3*5/12=5/36
Pr(A and C)= 1/4 * 5/12=5/48

Pr(None)=1- sum of the three other choices above.