Odds and Betting Calculation

Question

I need a formula to convert betting results into odds, I am trying to work out if a tipster is profitable or not.

For example if the average odds were 5.0 and the tipster was correct 1 in 5 times that would mean they would be losing (assuming the same stake) over time.

But if they predicted the outcome 1 in 4 times they would break even and 1 in 3 times would mean a profit.

How can I show this using a mathematical formula? I think I need to convert the "1 in 5" to decimal odds to compare but not sure how.

(Decimal odds of 5.0 is 4/1)

Answer 1

Suppose you are correct $1$ in $n$ times (and are wrong otherwise). When you are correct you receive $w$ and if you are wrong you receive $l$. Are you interested in what you receive on average?

If so, the probability of receiving $w$ is $\frac{1}{n}$ and the probability of receiving $l$ is $1-\frac{1}{n}=\frac{n-1}{n}$. On average you then receive $\frac{w}{n}+\frac{l(n-1)}{n}$.

Even simple way to see this is not to think about probabilities but number of times you receive $w$ or $l$. Out of $n$ guesses you are correct once and receive $w$ and are wrong $n-1$ times and receive $l$. Over the $n$ guesses you thus receive $w+l(n-1)$. Average over one guess is then $\frac{w+l(n-1)}{n}$.