I have $2$ jars, one of which is chosen randomly ($.5$ chance)
jar 1: $8$ blue marbles (b), $4$ red marbles (r)
jar 2: $15$ blue marbles, $3$ red marbles
In a situation WITH replacement I know the probability of the first pick being a blue marble given that the second pick (from the same jar as the first pick) is a red marble =
$$ P(p_1 = b \;|\; p_2 = r) = P(p_1=b \;|\; jar=1)P(jar=1 \;|\; p_2=r) + P(p_1=b \;|\; jar=2)P(jar=2 \;|\; p_2=r) $$
But what does the same probability WITHOUT replacement look like? I know this might be an easy question but I have tried a lot and nothing seems to add up. Thank in advance!