Let's say I play a variation of Black Jack against one opponent where the rules are as follows.
Both me and my opponent receive 2 random cards out of the standard 52 cards deck. The opponent looks at his cards. If he finds them strong enough, he can bet any amount of money to see my cards. If he finds that the combination of his cards beats combination of my cards(in terms of Black Jack) then he exposes his cards and I pay him 1 : 1.
I have a huge amount of data that show opponents' cards when we went to showdown. What I want to know is the probability of my opponent to bet the money when he has let's say a pair of sevens.
W(77) = P(betting | has pair of sevens) [1]
After an opponent bet the money and I exposed my cards he can either go to the showdown or muck his cards - these are two mutually exclusive events. The data I possess only show the former case when he went to showdown(WTSD).
Another piece of data I have is the overall probability of showdown event - P(WTSD) = 0.5 and overall probability of betting P(betting)
My idea is to rewrite the equation [1] to:
W(77) = P(betting AND WTSD | has pair of sevens) + P (betting AND NOT WTSD | has pair of sevens) [2]
then the first part of the expression on the right side can be calculated by using Bayes rule:
P(betting AND WTSD | has pair of sevens) = P(has pair of sevens | betting AND WTSD) * P(betting AND WTSD) / P(has pair of sevens). [3]
and I have all the data required to evaluate that expression.
and since P(WTSD) = 0.5 it seems that both parts on the right side of the equation [2] are equal and I can simply double the result of [3] to get the final number.
Does that make sense from the probability theory standpoint?
okay, I figured it all out, that's actually pretty, pretty simple
W(77) = P(betting | 77) = P(77 | betting ) * P(betting) / P (77) = (P(77 | wtsd) * P(wtsd) + P(77 | not wtsd) * P(not wtsd)) * P(betting) / P(77)