Find the PMF of variable using information from an other variable?

Question

There are 2 people in a city. Person A is moving with probability 1/4 to city 1, with probability 1/2 to city 2, with probability 1/4 to city 3. I create the probability mass function (PMF) of A: $$P_{A}(A) = \begin{cases} {\frac{1}{4}} &\quad\text{if }A =1\\ {\frac{1}{2}} &\quad\text{if }A=2\\ {\frac{1}{4}} &\quad\text{if }A=3 \\ \end{cases}$$

Person B follows person A in the same city with probability 1/2, otherwise with the same probability is moving to the other 2 cities.

What can we say about the PMF of person B?

Answer 1

Firstly notice that the probabilities of $A$ and $B$ going to different cities is given as, $$P(B=i|A=j) = P(B=j) = \frac{1}{4}$$

For a given city $i$, $$P(B=i) = \sum_{j}P(B=i|A=j)P(A=j)$$ Hence for $i=1$ $$P(B=1) = P(B=1|A=1)P(A=1)+ P(B=1|A=2)P(A=2)+ P(B=1|A=3)P(A=3)$$ $$P(B=1) = \frac{1}{2}\frac{1}{4}+ \frac{1}{4}\frac{1}{2}+ \frac{1}{4}\frac{1}{4}$$ And solve for other i's similarly.

Answer 2

You have been given information on the conditional probability of $B$ for given $A$.

Use the Law of Total Probability to evaluate the marginal probability for $B$.