You pass a street performer who offers you the following gambling deal:
You have 1/3 chance of winning 3 USD and 2/3 chance of losing 2 USD. However you may only play one game.
The street performer plays many independent games and therefore on average is likely to gain. The passerby, however, plays only one game and thus the law of large of numbers does not apply to him/her. Thus, the probability quoted by the street performer is of questionable relevance.
Should the passerby play the game? How many games must the passerby play before he/her is assured to a certain degree, to lose overall?
The way I thought about is that we have three parameters, $n$, the number of games. $P^*$ is a "Bernoulli-like" distribution where the probability of gaining 3 USD is 1/3 and losing 2 USD is 2/3. i.e. $P(X=3)=\frac{1}{3}=1-P(X=-2)$. And $q$ is the level of "certainty" that the I will gain from this interaction. I believe there are two degrees of freedom, e.g. our special case we know $P^*$ and we have $n=1$, and we ask "what is $q$?", or, "should I play?".