Today, Alice is playing games-mistress with her friend Marcus. To win the game, Marcus must pick two marbles of the same colour.

At the start, there are 6 marbles placed inside a small bowl. They are identical to the touch but 3 are blue and 3 yellow. Two empty pouches lie next to the bowl. The player is then blindfolded, the marbles randomised, and he then picks 3 marbles and places them inside one pouch, with the other 3 marbles going into the other pouch. He is then asked to select one pouch and pick just one marble from that pouch and put it inside the other one. At that point, one pouch will have 2 marbles and the other 4 marbles and the blindfolds can be removed. Marcus cannot see inside each pouch but he can feel which one has 2 and which 4 marbles. Lastly, he selects 2 marbles from just one chosen pouch. If the two colours are the same, he wins, if they are different, he loses.

The question is whether it is better to select the pouch with 2 marbles or 4 marbles? Does it make a difference, and if so, what are the relative probabilities to win?

The game is then repeated, with Alice being the player and Marcus the games-master. The whole process described above is merely to ensure that the player is confident that the game is random and not manipulated by a potentially unscrupulous opponent!

There is a follow-up to this but I shall post it separately.

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