Suppose we play a game with 50-50 probability, let's say even or odd, red or black. The fair odds should be 2.0 for each selection.

Now suppose we have a bookie who can't really calculate the real probability and he offers us new odds and also with no commission, that means 0% commission.

He offers us 1.5 -3.00. Is it worth taking the new odds? Yes it is, since 0.5*1.5 + 0.5*3 = 2.25. That means that we get about +12.5% on average, on our bet.

On the other hand, what happens if the real odds were 1.41-3.45 and the bookie offered us 1.49-3.03? Should we bet? Of course not, because 0.71*1.49 + 0.29*3.03 = 193.66, so we lose in average 3.5% of our bet.

Let's come in real world now. Supposing that a bookie almost always under or overestimates a result of a game with two possible outcomes, is it worth betting or not, just by choosing one of the 2 outcomes by luck?

I wrote 30 lines of code to simulate this. So I have real odds, under or overestimated odds by 10%, player's selection and an outcome. Of course, the outcome relies on the real odds, that means that if I have a set of odds 1.5-3.0 that means I have 100/1.5 = 0.66 for the first and 0.34 for the second outcome.

After running this for multiple times, 1 million everytime, I get a surprising result. That the bet should be +5% returned in average.

Can someone explain this to me mathematically ? Am I missing something maybe?

This post has been edited by **Larry71**: 07 September 2019 - 09:40 AM