Horse Racing Odds in Statistics I have a math project to do in statistics and probability. I am given a table of data related to horse racing biding which contain Odds of a Favorite Winning, if the favorite won etc.
So I have 300 races and the bookies odds of the favorite winning right, they are in form of:


*

*4/1

*9/4

*7/4

*11/4

*9/4


So now I have the following question:


*

*1) Calculate the average odds of favorite winning.

*2) What is the probability of a favorite winning.


I can't seem to understand how to calculate the 1st question. My approach was to add up the Odds like so: (in the given 5 samples)
4/1+
9/4+
7/4+
11/4+
9/4=
............
40/17  
which I divide both side by 5 (since there are 5 races in this sample), thus the average odds of the given samples is 8/3.4 (rounded up gives 8/3). Is that correct? I was told if you convert the odds to probability and then add up the mum of them and then divide by number of races you will get a skewed result. And correct me if I'm wrong but calculating the odds is for example -> given 11/4, the odds of happening are 11/(4+11) = 0.73 == 73%
Now for the 2nd question what I simply did is I took the number of wins and divided that by number of races which is basically relative frequency approach.. P(E) = # of desired outcomes / # of repetitions of the experiment.. which in this case relates to P(E) = # of wins / # of races, isn't that correct?
 A: Your approach is basically sound but I think you are confusing odds with probabilities.
Odds of 4:1 means that the book maker is prepared to give you £4 if the horse wins provided you give him £1 if it doesn't.  Typically the book maker will take your £1 up front and return it to you together with the £4 (making £5 in total) if you win but it can be just an agreement with no money changing hands until the result is known.
This means the horse has a $\frac{1}{5}$ chance of winning. So no looking at probabilities instead
$$
\begin{array}{c|c}
Odds & Probability \\
\hline
4 : 1 & \frac{1}{5} \\
9 : 4 & \frac{4}{13} \\
7 : 4 & \frac{4}{11} \\
11: 4 & \frac{4}{15} \\
9 : 4 & \frac{4}{13} \\ 
\end{array}
$$ 
Adding these up and dividing by 5 the average probability of a horse winning is $\frac{3101}{10725}$ making the odds 7620 : 3101
Also note the odds do not directly represent the real chances of horse winning but a perceived chance and the odds may go up or down depending on how much money has been bet.
So the real probaility of the favourite winning are just
$$
\frac{\text{The number of races won by the favourite}}{\text{total number of races}}
$$  
