Rank odds without converting

Question

Brazil are currently playing Mexico, and at the start of the game Brazil were 2/5 to win. As it's the 38th minute and still 0-0, their odds have changed to 8/15.

Now, if I'm not wrong that represents implied probabilities of 71.4% and 65.2% respectively.

What I'm wondering, is whether there's an easy way of working out which odds are better without having to convert them into percentages?

Answer 1

To compare two fractions $\frac ab$ and $\frac cd$, you can compute $ad-bc$. If it is greater than zero, $\frac ab \gt \frac cd$. If it is less, $\frac ab \lt \frac cd$ You are really just putting them over a common denominator.

Answer 2

I'm not sure if this is what you mean, but if you convert them to fractions, which does not involve a calculator, you get $5/7$ and $15/23$ respectively. If you multiply $5/7$ by $3/3$, you get $15/21$, which is greater than $15/23$, and so better odds.

Answer 3

Odds against of a to b is the same as a/b to 1 So 2-5 is 0.4 and 8-15 is 0.53 or just notice it is > 0.5 so it is larger than 0.4 . Higher "odds against" implies a lower prob of winning.

Answer 4

Or, try a "double division": $$\frac{\frac{2}{5}}{\frac{8}{15}}=\frac{15(2)}{8(5)}=\frac{30}{40}<1 $$