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

• 0
• 47/144
• 1/2
• 97/144
• 7/9

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: C

Let x be the probability of choosing A and B, but not C, y the probability of choosing A and C, but not B, z the probability of choosing B and C, but not A.

We want to find [math]w = 1 − ( x + y + z )[/math]. We have

[[math]] x + y = 1/4, \, x + z = 1/3, \, y +z = 5/12 [[/math]]

Adding these three equations gives

[[math]] \begin{align*} ( x + y) + ( x + z) + ( y + z) &= 1/4 + 1/3 + 5/12 \\ 2( x + y + z) &= 1 \\ x+ y+ z &= 1/2 \\ w &= 1-(x+y+z) = 1 -1/2 = 1/2 \end{align*} [[/math]]

Alternatively the three equations can be solved to give [math]x = 1/12[/math], [math]y = 1/6[/math], [math]z =1/4[/math] again leading to

[[math]] w = 1 - (1/12 + 1/6 + 1/4) = 1/2. [[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.