Don't take this the wrong way ... probabilities are easy to get wrong ... I'm only having the conversation with you because you're getting it completely wrong, and yet you appear to want to learn.
You say:
P(blue_head OR red_head) = 1
you said that you're retrying until this happens
normally this would be 3/4
This is an incorrect application of Bayes' Theorem. The whole point of taking P(X|Y) isn't that P(Y) can be then taken as being 1. P(Y) remains the probability of Y. The point is that when we then only consider those events where Y occurs, then we have conditioned our probabilities, and the formula gives us what we want. Specifically, relative to Y, the probability of X changes.
Using "BH" for "Blue coin shows Head" and "RH" for "Red coin shows Head" we have:
P(BH and RH | BH or RH)
= P(BH and RH and (BH or RH)) / P(BH or RH)
= P(BH and RH) / P(BH or RH)
= 0.25 / 0.75
= 1/3
You then go on to say:
P(BH OR RH) = P(BH) + P(RH) - P(BH) * P(RH) = 1
Taking just the second part of this:
P(BH) + P(RH) - P(BH) * P(RH) = 1
and adding P(BH) * P(RH) to both sides we get:
P(BH) + P(RH) = 1 + P(BH) * P(RH)
Since by symmetry P(RH)=P(BH), and letting x=P(BH)=P(RH) this simplifies to
x + x = 1 + x^2
The only solution to that is x=1, so P(BH)=P(RH)=1
Your math is clearly screwed at this point.
So I've given you the correct interpretation, I've shown you where your calculations are wrong, and I've explained your incorrect use of Bayes' Formula.
Anyway ...
Don't take this the wrong way ... probabilities are easy to get wrong ... I'm only having the conversation with you because you're getting it completely wrong, and yet you appear to want to learn.
You say:
This is an incorrect application of Bayes' Theorem. The whole point of taking P(X|Y) isn't that P(Y) can be then taken as being 1. P(Y) remains the probability of Y. The point is that when we then only consider those events where Y occurs, then we have conditioned our probabilities, and the formula gives us what we want. Specifically, relative to Y, the probability of X changes.Using "BH" for "Blue coin shows Head" and "RH" for "Red coin shows Head" we have:
You then go on to say: Taking just the second part of this: and adding P(BH) * P(RH) to both sides we get: Since by symmetry P(RH)=P(BH), and letting x=P(BH)=P(RH) this simplifies to The only solution to that is x=1, so P(BH)=P(RH)=1Your math is clearly screwed at this point.
So I've given you the correct interpretation, I've shown you where your calculations are wrong, and I've explained your incorrect use of Bayes' Formula.
Twice.