2 probabilities

Question

The end result should only be 1 of 2 (green or blue).

Scenario 1 : A+B has 55% probability of equalling green.

Scenario 2 : C+D has 72% probability of equalling green.

Question : can the 2 scenarios relate to each other and provide a better green probability, if so, how to calculate it ?

( what's the overall probability of each color separately )

Answer 1

Yes--let P(A+B) be the probability of A+B being green, and P(C+D) be the probability of C+D being green.

If you combine A+B and C+D, you will get the probability that A+B &/or C+D will be green, which equals:

P( A+B &/or C+D ) = P(A+B) + P(C+D) - P( A+B & C+D ), since we would double count the probability

that both A+B and C+D are green, so P( A+B &/or C+D ) = P(A+B) + P(C+D) - P(A+B) * P(C+D) =

55% + 72% - 55% * 72% = 127% - 5*11*72% / 100 = 127% - 792% / 20 = 127% - 39.6% = 87.4%.

Now, finally we can also calculate the probability that both A+B and C+D are blue to be the following:

100% - P( A+B &/or C+D ) = 100% - 87.4% = 12.6%.