# Probability in terms of odds for and against the event

CFA level I / Quantitative Methods: Basic Concepts / Probability Concepts / Probability in terms of odds for and against the event

The odds of an event is the probability of occurrence of an event to the probability of not occurrence of the event. If the odds in favor of market going up tomorrow is 2 to 5, that means that the market is expected to go up two times and not to go up five times. So, the probability of the market going up will be equal to 2/7.

If the probability of an event E is P(E), then the odds for that event equals P(E)/[1-P(E)].

If the odds for occurrence for an event is x:y, then the probability of occurrence of that event equals x/(x + y).

If the odds against the occurrence of an event is x:y, then the odds in favor of that event would be y:x and the probability of the occurrence of that event would be y/(y + x)

Profiting from inconsistent probabilities: Suppose there are two companies A and B operating in the same industry and a decision of banning imports would help both the companies equally. Suppose the market price of company A indicates a probability of 25 percent probability of import ban and the market price of company B indicates 50 percent probability. Now, if 25 percent figure is correct then company A is fairly valued, and company B is overvalued. If 50 percent figure is correct, then company B is fairly valued, and company A is undervalued. In both scenarios, the company A is a better investment. So, an investor would be better off buying company A. A more aggressive approach would be to buy company A and simultaneously selling company B. This type of trade is called pair arbitrage. This profit opportunity from arbitrage has arrived due to inconsistent probabilities, and it is also referred to as the Dutch Book Theorem.

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