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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 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?

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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?

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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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