The probability of a probability to occur

Question

I was wondering for so long about a problem that i haven't found a solution for yet, That is. Let's suppose that we have an event (E) that has a chance of 50% to occur (and 50% not to occur). Let's take P(E) (50% that the event (E) will occur). And suppose that we know some True variables in this section (P(E)) are existing and thus. P(E)>=P(E)[bar]. {For more details. We have for example

• an event that:player1 wins against player2 Let's call this event (G) The probability of (G) with respect to P1 and P2 is, P(G)=0.5=50%.

• in fact. We know that the P1 has trained alot and P2 has (for example) a broken leg. Certainly. P(G) will be for sure more than P(G)[bar]. } P(x)[bar] is the inverse of P(X) ... do you think i am right ? If so. Does it have a formula to calculate it (the real probability i call it) ?

Answer 1

You're describing a conditional probability. We would say that we are seeking the "probability of event G given P1 has practiced hard and P2 has a broken leg." Mathematically, this conditional probability is written $\mathbb{P}(G|\mbox{P1 practised and P2 has a broken leg})$. This is generally computable by the definition,

$\mathbb{P}(A|B)=\frac{\mathbb{P}(A,B)}{\mathbb{P}(B)}$,

Although you may not know the values in the above equation in which case you can't compute the probability and should probably try to estimate it from real life observations.