In a certain group of cancer patients, each patient's cancer is classified in exactly one of the following five stages: stage 0, stage 1, stage 2, stage 3, or stage 4.

1. 75% of the patients in the group have stage 2 or lower.
2. 80% of the patients in the group have stage 1 or higher.
3. 80% of the patients in the group have stage 0, 1, 3, or 4.

One patient from the group is randomly selected.

Calculate the probability that the selected patient's cancer is stage 1.

• 0.20
• 0.25
• 0.35
• 0.48
• 0.65

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

Solution: C

Let [math]p_i[/math] represent the probability that the patient's cancer is in stage i, for i = 0, 1, 2, 3, or 4. The probabilities must sum to 1. That fact and the three facts given the question produce the following equations.

[[math]] \begin{align*} p_0 + p_1 + p_2 + p_3 + p_4 =1 \\ p_0 + p_1 + p_2 = 0.75 \\ p_1 + p_2 + p_3 + p_4 = 0.8 \\ p_0 + p_1 + p_3 + p_4 = 0.8 \end{align*} [[/math]]

Therefore, we have

[[math]] \begin{align*} p_0 =( p_0 + p_1 + p_2 + p_3 + p_4 ) − ( p_1 + p_2 + p_3 + p_4 ) = 1-0.8 = 0.2\\ p_2 =( p_0 + p_1 + p_2 + p_3 + p_4 ) − ( p_0 + p_1 + p_3 + p_4 ) =1 − 0.8 =0.2 . \\ p_1 = ( p_0 + p_1 + p_2 ) − p_0 − p_2 = 0.75 − 0.2 − 0.2 = 0.35. \end{align*} [[/math]]

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