Question

The victims of a certain disease being treated at Wake Medical Center are classified annually as follows: cured, in temporary remission, sick, or dead from the disease. Once a patient is cured, he is permanently immune. Each year, those in remission get sick again with probability 0.08, are cured with probability 0.27, die with probability 0.05, and stay in remission with probability 0.6. Those who are sick are cured with probability 0.05, die with probability 0.2, go into remission with probability 0.4, and remain sick with probability 0.35. Find the transition matrix and do the calculations necessary to answer these questions. (a) If a patient is now in remission, what is the probability he is still alive in 2 years? (Hint: In which of the states is the patient still alive?) ] (b) If a patient is now in remission, what is the probability he dies within 3 years?

Answer 1

The transition matrix for the disease shows the probability of a patient moving from one state to another in one year. For example, the probability of a patient who is now in remission getting sick again in one year is 0.08. The probability of a patient who is now in remission staying in remission in one year is 0.6.

The probability of a patient who is now in remission being still alive in 2 years is the sum of the probabilities of the patient being in each of the alive states in 2 years. These states are cured and remission.

The probability of the patient being cured in 2 years is 0.27 * 0.27 = 0.0729. This is because the probability of the patient being cured in one year is 0.27, and the probability of the patient being cured in two years is the probability of the patient being cured in one year times the probability of the patient being cured in the second year.

The probability of the patient being in remission in 2 years is 0.6 * 0.6 * 0.6 = 0.216. This is because the probability of the patient staying in remission in one year is 0.6, and the probability of the patient staying in remission in two years is the probability of the patient staying in remission in one year times the probability of the patient staying in remission in the second year.

Therefore, the probability of the patient being still alive in 2 years is 0.0729 + 0.216 = 0.2889.

The probability of a patient who is now in remission dying within 3 years is 1 - 0.2889 = 0.7111. This is because the probability of the patient dying within 3 years is the complement of the probability of the patient being alive in 2 years.