Reading various betting forum I came across different threads claiming betting multiple is worse than betting on single events.
Could you explain why?
[Clairification for the ones not familiar with betting: Betting on a single event: predict the outcome of a single match. Betting on multiple events: predict the outcome of multiple matches, you win only if you guess correctly the outcome of all the events, even if you only make one mistake you win nothing.]
I think that the key point is the commission taken by bookmakers on each match (they take something like 5% on every bet).
For example if you bet 100dollars on a tennis match where each player has P(win) = 0.5, You will win something like 195 instead of 200, due to commissions.
Assuming the outcomes of all the matches are independent, with $p_k$ being the probability of a favorable outcome of the $k^{th}$ match, for matches numbered $1$ to $n$. Then probability of favourable outcomes occurring in all matched will be:$$P_{\text{all}}=\prod\limits_{k=1}^n p_k$$
Since $0\leq p_j\leq 1$ for all $j$, then it follows that: $p_k\geq \prod\limits_{j=1}^n p_j$ for all k.