# laplace law of succession paradox

(I am a newcomer in bayesian probability... so please be kind if this is very naive!)

Laplace law of succession: (r+1)/(n+2) for a binary next event

Jaynes ("Probability Theory", p 571) appears to extend it to: (r+1)/(n+k) when the event has k possibilities.

Now suppose: you draw 4 yellow balls. Assume that you there are 1'000 color possibilities (perhaps because your colorimeter has only 3 digits).

Then the probability that the next ball is yellow would be: (4+1)/(4+1000) about 0.5%

But you can also view it this way: the ball can be either yellow or NOT yellow (any other of the 1'000 colors). Therefore since you have a binary event: yellow vs non-yellow the probability of yellow would be (4+1)/(4+2) = 5/6

This is quite different from the previous number (you could then say that the probability of a specific non-yellow color is (5/6) / 999.

So what is really the probability that the next ball is yellow: 0.5% or 5/6 ?

• You completely changed the probabilities by changing your description to "yellow or not yellow." Contemplate, then, how and why an English description ought (or ought not) have any bearing on your probability model.
– whuber
Commented Jul 10 at 19:41
• I could also say: yellow OR (one of the 999 other colors). I don't see how this solves the paradox Commented Jul 10 at 19:53
• There is no paradox, because there is no law of probability that asserts the probabilities of any pair of binary choices must be equal.
– whuber
Commented Jul 10 at 20:51
• I don't see the relation with the question asked Commented Jul 10 at 20:56
• The $n$ in the Laplace law of succession is the count of past observations, not potential future observations, and not the number of possibilities for each observation. You are misusing it here. Commented Jul 10 at 21:00