# Guess the box

Today we’re going to play a bit with the Bayesian statistics using a small game analysis.

We have two players, Bob and Alice, both skilled players and statisticians. They have 100 white and 100 black balls and 2 boxes, left and right.

First, Bob splits all the balls into boxes so that there are 100 balls on the left and 100 on the right, and tells Alice the number of black balls he decided to put on left. Then he randomly picks a box without Alice knowing, and pulls a single ball out.

Based on the colour of that ball, Alice has to guess the box that was chosen.

Bob picked the box and pulled a white ball out, telling Alice there are 99 black balls on the left. After this information, Alice had no preference over left or right and tossed a coin to choose an answer.

Bob suspects her of cheating, is he right?

Given that the boxes are, [LEFT 1white 99black] and [RIGHT 99white 1black], it is strange that Alice guessed, what could have been her reasoning than?

Alice needs to decide the probability of a box given that white ball was chosen, that is for left box, P(L|W). Using Bayes theorem, we get a simple equation.

Since she knows the distribution of colours, she also knows the probabilities of pulling white ball from each box. Assuming we don’t know her prior belief for now, what’s her posterior?

Alice, as a statistician, should come up with this too, yet, based on her own result she wasn’t able to conclude the side to pick and rather chose to toss a coin.

We may set her posterior equal to 1/2 as she was indecisive, and solve for prior. That’s a simple equation P(L|W) = 0.5, which leads to P(L) = 99/100.

This leads us to idea, that before knowing the ball colour, Alice was 99% sure that Bob chose the left box rather than right.

Was she cheating? Is in fact Bob skilled statistician when he chose this particular colour distribution?

See you next time.

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