Hi,

I am playing a game. I win with an 88% chance and lose with 12% probability.

If I have won 20 games in a row what is the probabilty that I will lose the next game.

The wording of your question implies that you wish to find \(P(G_{21}=L|G_1=W , G_2=W , ... , G_{20}=W)\), the probability of losing the 21st game,

**conditional** on having won all of the first 20. If the probability of winning/losing is the same for each game, and all games are mutually independent, then the probability of losing the next game after having won the previous 20 is still 12%, since independence means that the chance of winning a game is not affected by the chance of winning another game.

That's a different question though than \(P(G_1=W , G_2=W , ... , G_{20}=W , G_{21}=L)\), the probability of the

**joint** event of 20 straight wins followed by one loss. This type of event is described by the geometric distribution (a special case of the negative binomial distribution). Again assuming independence and constant win/loss probability, then

\(P(G_1=W, G_2=W , ... , G_{20}=W , G_{21}=L) = P(G=W)^{20}P(G=L) = (0.88^{20})(0.12)\).

If there is dependence among the game outcomes, then more information would be needed.