A game is 90% successful in determining a good player. However, 2% of the time it allows a bad player to pass. Three people out of a hundred are good players. What is the probability of being a good player given the game has been passed?
I was thinking this is a bayesian problem: Probability of being a good player, given the game has been passed:
P(Good player| Game passed) = P(Game passed| Good Player)*P(Good Player)/P(Game passed)
Where:
P(Good Player) = 3/100 = 0.03
P(Game passed) = 1
P(Game passed | Good Player) = 0.9*(1-0.02) = 0.882
So:
P(Good player| Game passed) = 0.882*0.03/1 = 0.02646
I was wondering if there was anything wrong in my thought process?
The relevant events shall be labelled $B$ for bad player, and $G$ for game passed.
We are told:
A game is 90% successful in determining a good player.
When the player is good, the game is passed with $90\%$ success rate.
$$\mathsf P(G\mid B^\complement)=0.90$$
However, 2% of the time it allows a bad player to pass.
When the player is bad, the game is passed with $2\%$ success rate.
$$\mathsf P(G\mid B)=0.02$$
Three people out of a hundred are good players.
$$\mathsf P(B^\complement)=0.03$$
Now you can find $\mathsf P(B^\complement\mid G)$ using Bayes' rule and the Law of Total Probability