Probability and Expectation Analysis

A man is drinking a beer in a pub, he feels boring. A lady comes to him, takes out two coins and says: “Let’s play a game. Let’s throw a coin each. And if both are heads, you win $3, if both are tails, you win $1. Otherwise I win $2”

So, put this in a table:pro

Lady\Men	Head	Tail
Head	        3	-2
Tail	       -2	 1

The positive numbers mean a gain for the man and the negative numbers mean a gain for the lady. The man aims to maximum the numbers while the lady is minimizing the numbers.

At a glance, each is 25% probability so it should be quite fair for each player. For example:

Both heads 1/4
Both tails 1/4
Head + Tail 1/4
Tail + Head 1/4

Expectation

The expectation is the probability of winning times the gain. Let’s assume for the man, the probability of heads is x and the probability of tails is thus 1-x. Similarly, let’s assume for the lady, the probability of heads is y and the corresponding tails is 1-y.

Thus the expectation for the man winning is

$tex_a16ee981080d7187ebfd6f5f95c59d4d Probability and Expectation Analysis math probability$

The man want’s to maximum the above equation. Let’s rearrange the equation to:

3xy + 1 + xy – x – y – 2(y – xy + x – xy)
3xy + 1 + xy – x – y – 2y + 2xy – 2x + 2xy
8xy – 3x – 3y + 1
y(8x – 3) – 3x + 1

So, this should be positive for the man, thus y(8x-3)-3x+1 > 0 and we should get:

(8x-3)y > 3x-1

So, if 8x – 3 > 0 which is x > 0.375, y > (3x-1)/(8x-3) (for man wining). And on the other hand,
when x < 0.375, y > (3x-1)/(8x-3) (for man winning).

If x=3/8 which means if the probability of the man’s throwing heads is 0.375, the expectation is -0.125 In the long term, the main is losing.

Let’s say that the lady knows x, then, she will adjust the y so that above expectation for the man is smaller than zero, which is indidated in below graph.