# Probability

Probability is found everywhere in quantitative finance because the financial markets are full of uncertainty. The ability to quantify that uncertainty and make predictions is key to landing a job in quantitative finance and succeeding in the industry.

This quiz and the two that follow it assume a working knowledge of the rules of probability, and are focused on highlighting the problem-solving skills needed to tackle challenging questions. For a more complete review of probability, check out the Probability course.

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Red and blue 20-sided dice numbered 1-20 are rolled. What is the probability that the number on the red die is greater than the number on the blue die?

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One clever way to approach the previous problem is through “symmetry.” Because the red and blue die are symmetric, the probability that the red die is greater than the blue die is the same as the probability that the blue die is greater than the red die. Since the dice are the same, and the probability of a non-tie is $$1-\frac{1}{20} = \frac{19}{20},$$ then by symmetry, the desired probability is $\frac{1}{2} \cdot \frac{19}{20} = \frac{19}{40}.$

The next three questions highlight problems which also make use of symmetry.

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Alice rolls a five sided die (numbered 1-5) and Bob rolls a ten sided die (numbered 1-10). Alice wins if her number is greater than Bob's, and Bob wins otherwise. What is the probability that Alice wins?

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Five fair coins are flipped. What is the probability that more than half of them are heads?

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Amy flips 4 coins, while Brad flips 5. What is the probability that Brad flips more heads than Amy?

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Another use of probability - especially in quantitative finance - is modeling the likelihood of change. For example, you can say that a stock will go up or down with some probability over some time interval. This type of model, when taken at very small time intervals, actually forms the basis of some more complex models in quantitative finance.

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Every day, a stock price either goes up 1% with probability 0.5 or down 1% with probability 0.5. After 4 days, what is the probability that the stock price is greater than where it started?

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The probability that at least one company is going to announce a takeover in the next hour is 84%. What is the probability that at least one announces in the next 30 minutes? You may assume that if $$I_{1}$$ and $$I_{2}$$ are two non-overlapping time intervals, the probability of no announcement in $$I_{1}$$ is independent of the probability of no announcement in $$I_{2} .$$

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