for the explanation for the dutch book theorem there is an example like… "suppose someone places a $100 bet on X at odds of 10 to 1 against X, and later he is able to place a $600 bet against X at odds of 1 to 1 against X. Whatever the outcome of , that person makes a riskless profit equal to $400 if X occurs or $500 if X doesn’t occur.

can you please explain how?

If X wins, the first bet pays off $1,000 and the second bet loses $600; net result: +$400.

If X loses, the first bet loses $100 and the second bet wins $600; net result: +$500.

Heads, you win; tails, you win.

thank you for the reply…but aren’t they ruling out the probability of losing or winning both of them? how does it become a riskless profit?

You cannot lose or win both: the first bet is that the Vikings will beat the Panthers today, and the second is that the Panthers will beat the Vikings today. You have two bets on a single outcome: one bet has to win, and the other has to lose.