Probability of Profitability Imagine I am betting on the outcome of flipping $n$ biased coins $C_i$. I know exactly what the probability of each coin landing on heads is $P(H)$ and exactly the probability of each coin landing on tails is $P(T)$. For every flip I can place one bet on either (but never both) outcome at odds with 5% value i.e. the probability implied by the odds is 5% lower than $P(H)$ or $P(T)$. Because I am getting value on each bet, I know that in each case my expected value is positive. If I choose to bet on $n$ coin flips, what is the probability that I will make a net loss? Cheers, Pete

If P(H) varies by coin, it will be hard to do better than adding up all the cases. If P(H) is the same for all coins and P(T)=1-P(H) you can use the binomial theorem. First calculate how many coins need to come heads to make a profit. For example, if P(H) = 0.5 you should win 1/.45 -1= 11/9 for heads (is this what you mean for 5% value? I could argue for various other figures, but let's use this) and lose 1 for tails. To break even, then, you need to get tails 11/20 of the time and to lose you need more tails than that. So the probability of loss in $n$ throws is $$\frac{1}{2^n}\sum_{k=0}^{\lceil \frac{9k}{20} \rceil -1} \binom{n}{k}$$ If the coin is not fair, you will need to bring the P(H) into the sum.