You have (a belief about) a hypothesis. See some evidence about it. How should this information update your prior
belief.
Instead of using the meaningless A and B, we use here H for hypothesis and E for evidence.
P(H) the probability of the hypothesis being true before any evidence.
The current belief about the hypothesis; the "prior".
P(E|H) the probability of seeing evidence if the hypothesis is true. The "likelihood".
P(E) the probability of seeing evidence.
P(H|E) the probability the hypothesis is true given some evidence, i.e. when E is true.
The new, updated belief about the hypothesis; the "posterior".
Bayes' formula:
P(H|E) = P(H)·P(E|H) / P(E)
P(E)=P(H)·P(E|H)+P(H̄)·P(E|H̄)
How often the hypothesis is true among the cases where the evidence is true , i.e. what proportion.
Box 1 has 2 red, 1 blue ball Box 2 has 1 red, 3 blue balls select a box, draw a ball. Sequence of dependent events: box, then ball P(B1)=P(B2)=1/2 P(red)=1/2*2/3+1/2*1/4 = 11/24=0.458333 the probability of seeing red P(blue)=1/2*1/3+1/2*3/4 = 13/24=541666 =P(~red) P(red|B1) = P(red∩B1)/P(B1) = 1/2*2/3 / 1/2 = 2/3 P(red|B2) = P(red∩B2)/P(B2) = 1/2*1/4 / 1/2 = 1/4 P(blue|B1) = P(blue∩B1)/P(B1) = 1/2*1/3 / 1/2 = 1/3 P(blue|B2) = P(blue∩B2)/P(B2) = 1/2*3/4 / 1/2 = 3/4 P(B1|red) = the color, the second step in the sequence, is the Evidence; box is the Hypothesis P(H)=P(B1)=0.5 your "prior" belief about boxes, 50-50 P(E|H)=P(red|B1)=.6666 the likelihood of seeing red if it's box 1 P(E|~H)=P(red|B2)=.25 or P(E)=P(B1)·P(red|B1)+P(B2)·P(red|B2)=1/2*2/3+1/2*1/4=0.458333 =0.7272=8/11 your "posterior" belief that it was box 1 whence the red P(B2|red) = P(B1|blue) = P(B2|blue) =