If you think there is no way to make a riskless profit, think again. This concept is known as *Dutch Book*. First, let me remind the definition of the probability stated as odds. Given a probability $P(E)$ of an event $E$, **odds for** $E$ are equal

$$ E=P(E)/[1-P(E)] . $$ Given odds for $E$ of **a:b**, the implied probability of $E$ is $a/(a+b)$. Reversely speaking, **odds against** $E$ are $$ E=[1-P(E)]/P(E). $$ Thus, given odds against $E$ of **a:b**, the implied probability of $E$ is $b/(a+b)$.

Now, suppose John places $\$100$ bet on $X$ at odds of 10:1 against $X$, and later he is able to place a $\$600$ bet against $X$ at odds of 1:1 against $X$. Whatever the outcome of $X$, that person makes a riskless profit equal to $\$400$ if $X$ occurs or $\$500$ if $X$ does not occur because the implied *probabilities are inconsistent*.

John is said to have made a **Dutch Book** in $X$.