Every year, $10 \%$ of the population of San Francisco moves to Sacramento, and $20 \%$ of the population of Sacramento moves to San Francisco. The current population of San Francisco is 10 million, and the current population of Sacramento is 4 million. Given that a person currently lives in Sacramento, what is the probability that s/he will live in San Francisco three years from now? 0.445 0.493 0.425 0.438 0.552

Step 1 ：The problem is asking for the probability that a person currently living in Sacramento will be living in San Francisco three years from now. This involves a transition of population between the two cities over a period of three years.

Step 2 ：We know that every year, 10% of the population of San Francisco moves to Sacramento, and 20% of the population of Sacramento moves to San Francisco.

Step 3 ：We can model this as a Markov chain, where the states represent the two cities, and the transition probabilities are given by the percentages of people moving from one city to the other each year.

Step 4 ：We can then calculate the probability of a person currently in Sacramento being in San Francisco three years from now by raising the transition matrix to the power of 3 (representing three years), and taking the appropriate entry.

Step 5 ：The transition matrix P is given by: \[ P = \begin{bmatrix} 0.9 & 0.2 \\ 0.1 & 0.8 \end{bmatrix} \]

Step 6 ：Raising the transition matrix to the power of 3, we get: \[ P_3 = \begin{bmatrix} 0.781 & 0.438 \\ 0.219 & 0.562 \end{bmatrix} \]

Step 7 ：The probability that a person currently living in Sacramento will be living in San Francisco three years from now is given by the entry in the second row and first column of the matrix P_3, which is 0.219.

Step 8 ：Final Answer: The probability that a person currently living in Sacramento will be living in San Francisco three years from now is approximately \(\boxed{0.219}\) or \(\boxed{21.9\%}\).