If two events a and b are independent and you know that p(a) = 0.65, what is the value of p(a | b)?

Let's think of an example which would better help visualize this situation. 

1) Someone exercises a lot so they can concentrate more on their homework (here running and concentration on homework are dependent events)

2) Someone exercises a lot because they wear a blue t-shirt (here, you can clearly see that wearing a blue shirt and exercising are not related. These events are independent).

Now concentrating on P(a | b) = Probability a occurred if b occurred. 

It does not matter if b occurred (just like it didn't matter that the person wore a blue shirt which meant that they exercised) for the outcome a to occur. 

Therefore, probability of P(a | b) = P(a) 

P(a | b) = 0.65 

Hope I helped :)  

Ais the answer for your question