
Odds vs Probability
A newspaper article preceeding the 1994 World Cup semifinal match between Italy and Bulgaria stated that "Itality is favored 1011 to beat Bulgaria, which is rated at 103 to reach the final". Suppose this means that the odds that Italy wins are 11/10 and the odds that Bulgaria wins are 3/10. Find the probability that each team wins, and comment.
Okay, so I know the transformation from Odds to Probability is given by:
Probability = Odds / 1 + Odds
and the transformation from Probabilities to Odds is given by:
Odds = Probability / 1  Probability
Plugging in, I get:
Probability Italy wins: 0.23
Probability Bulgaria wins: 0.52
Comment: So, from the question, it sounds like Bulgaria is less likely to win given the odds. However, if we calculate the probabilities, its clear that Bulgaria actually has a better shot at winning.
Is this how you would approach the question? I found it a bit easy, I'm afraid I've missed something or interpreted it in a different way.
Cheers

Hi GoodRobot,
you've miscalculated...
Italy at bookmaker odds of $\displaystyle \frac{10}{11}$ is $\displaystyle Odds_{win}=\frac{11}{10}\ \Rightarrow\ P_{win}=\frac{\frac{11}{10}}{\frac{11}{10}+\frac{1 0}{10}}=\frac{11}{11+10}=\frac{11}{21}$
Converting back...
$\displaystyle Odds_{win}=\frac{P}{1P}=\frac{\frac{11}{21}}{\frac{21}{21}\frac{11}{21}}$
Similarly for Bulgaria

Ahh, I missed that.
Thank you :)(Smirk)