Probability and statistics are the foundation of most quantitative financial models. They allow us to interpret large amounts of data and model future outcomes in a way that humans could not do on their own.

A biotechnology stock is currently trading at $12, and the company is releasing the results of a drug test tomorrow. An analyst tells you that if the test was successful, the stock should rise to $20; otherwise, it will fall to $8.

If the current stock price is equal to its future expected value, what is the probability that the drug test will be announced as successful as implied by the analyst?

Over time, you have found that the price of a certain asset seems to follow a Markov model in which it will increase or decrease minute-to-minute according to the model above. For example, if its price increases one minute, it is 80% likely to increase again in the next minute. In the long run, in what portion of all minutes does the stock price increase?

**Hint:** If it increases with probability $p,$ it decreases with probability $1-p.$ What value of $p$ would provide a steady state in this chain?

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%?