Question

A company sometimes stops payments of quarterly dividends. If the company pays the quarterly dividend, the probability that the next one will be paid is 0.7. If the company stops the quarterly dividend, the probability that the next quarterly dividend will not be paid is 0.5. Then the probability (rounded off to three decimal places) that the company will not pay quarterly dividend in the long run is ________

Hint:

For such problems, you can model the system as a Markov chain and compute the steady-state probability of the system.

Answer 1

Correct Answer: 0.375

Let \( p \) be the probability that the company pays the quarterly dividend, and \( 1 - p \) be the probability that the company does not pay it. The problem gives us the following conditions: - If the company pays the dividend, the probability that the next one will be paid is 0.7. Thus, \( p = 0.7 \). - If the company does not pay the dividend, the probability that the next dividend will not be paid is 0.5. Hence, \( 1 - p = 0.5 \). Now, in the long run, the total probability that the company does not pay the dividend can be calculated as follows: \[ P(\text{no payment}) = (1 - p) \times (1 - p) + p \times (1 - 0.7) = 0.5 \times 0.5 + 0.7 \times 0.3 = 0.25 + 0.21 = 0.375. \] Thus, the probability that the company will not pay the quarterly dividend in the long run is \( \boxed{0.375} \).