Question

There are three cards: one has both sides red, one has both sides green and one has a red side and a green side.

One card is randomly chosen and one side of it is shown to a player. The player’s goal is to guess the other side’s color. The player wins $1 if he guesses right and loses $1 if he guesses wrong.

Is there a winning long-term strategy? Find thr winning strategy and the probability of wining if following the stratefy.Use Bayes's Law.

Answer 1

Let us start with an example,

Probability of choosing green is 3/6 i.e. 1/2.

After choosing green, you have two cards left, one having green on both sides and one having green and red on other side.

After a green is choosen, probability of having green on other side is 2/3.

As per Bayes theorem,

P(A/B) = P(B/A) * P(A) / P(B)

P(R/G) = P(G/R) * P(R) / P(G)

= (1/3 * 1/2 ) / (1/2) = 1/3

Hence probability of choosing red given shown a green card is 1/3.

For long term strategy, it better to choose same color card after a color is shown to you. The chances are winning are 2/3.