If the stock is perfectly priced according 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 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 value increases one minute, it is 80% likely to increase again in the next minute. In the long run, in what percentage of minutes does the stock 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 arbitrage be executed before the probability of at least one failure exceeds 50%?

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