For a stocks trader, given the probability of having a profitable day, how to determine probability of having a profitable year? Lets assume that, for a particular stocks trader, the probability that he makes profits on any given day is 0.52. So there is 52% chance that his day will be profitable, and a 48% chance that he will end up in losses. What is the probability that the trader is profitable over a given month or a year? Probability of having a profitable day = 0.52 Probability of having a profitable month = ? (Assuming 20 trading days in a month) Probability of having a profitable year = ? (Assuming 240 trading days in a year) Edited: I forgot to add, lets say that the profits and losses are of the same size.

If loss=profit, then if a broker works $m$ days a month (for simplicity assume $m \in 2 \mathbb{Z}$), then what he needs to do to break even is have at least $\frac{m}{2}+1$ profitable days. The probability of this (I use $p$ for probability of winning and $q$ for the probability of losing, $S$ is the event that profit exceeds loss over a month): $$ P(S)=\sum_{k=\frac{m}{2}+1}^{m}\binom{m}{k}p^{k}q^{m-k} $$ Can you extend this idea to the whole year?