# Probability and statistics in Quantitative Finance

Certain types of traders attempt to repeatedly buy and sell the same asset for a profit over a short time period, such as high-frequency “market makers”. For example, if you can repeatedly sell a stock for \$8.50 and buy it for \$8.49, you will make \\$0.01 each time. This is known as arbitrage.

If this transaction succeeds with probability 99%, about how many times can this transaction be executed before the probability of at least one failure exceeds 50%?

If you take $$N$$ attempts, then the probability of at least one (or more) failures is the complementary probability of never failing on those $$N$$ attempts:
$$p=1-p_{pass}^N$$
Solving for $$p=50\%$$, you need to evaluate $$p_{pass}^{N}=0.5$$, where $$p_{pass}=0.99$$, getting that $$N\geq 69$$.