Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have a dataset made of couples $(n_i,v_i)$ where $n_i$ denotes the number of times a game has been played, and $v_i$ the number of victories at the $i-$th day.

What is the best way to evaluate the probability $P$ of winning the game? (We can assume that winning the game does not depend on time).

My first thought is to evaluate $P$ as the total number of victories over the number of games, i.e. $$P = \frac{\sum_{i=1}^t v_i}{\sum_{i=1}^t n_i} $$

I then had the doubt though that I could evaluate $p_i = v_i / n_i$ and define $P$ as the mean of the individual probabilities: $$ P = \frac{1}{t} \sum_{i=1}^t p_i $$.

Somehow I feel this second approach is wrong, but can't entirely understand why.

How would you evaluate $P$ and why? Can you give me some links explaining how to evaluate reliable statistics?

share|cite|improve this question

The answer may depend on whart you know beforehand about that probability $p$. What we can says, is: If the probability is $p$ then the probability of observing $v=\sum v_i$ victories within $n=\sum n_i$ games is $${n\choose v}p^v(1-p^{n-v}) $$ and this expression is maximal when $p=\frac v n$. Thus if we have no a priori knowledga about $p$ (i.e. consider each value $\in[0,1]$ equally likely), then $p=\frac vn$ is the best guess.

You can also do this on a basis, but then your adjustment after day $i$ must take into account that you do have some a priori knowledge. You need to apply Bayes theory and this will in the end lead to exactly the same result.

share|cite|improve this answer
Thanks Hagen, but I can't really follow your reasoning (it is a long time since I applied probability last time). Can you please send me some reference? Why would maximizing p be the best guess? Anyway yes, I assume that p is uniform in [0,1]. I think this is quite standard job for statisticians, and I am looking at the standard way to do this. Also I don't understand how a priori knowledge would influence the result, since $p_i$ doesn't depend on previous times... – lucacerone Oct 6 '12 at 12:11

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.