Level 1 CFA® Exam:
Bayes' Formula

Last updated: July 13, 2022

Bayes' Formula for Level 1 CFA Candidates

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Bayes' formula will allow you to update probability based on new information. We may say that Bayes' formula works like an inversed probability.

BAYES' FORMULA: The updated probability of an event given the new information (posterior probability) is equal to the probability of the new information given event divided by the unconditional probability of the new information times prior probability of event.

Here's the formula for Bayes' theorem:

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\(P(E|I) = \frac{P(I|E)}{P(I)}\times P(E)\)

  • \(P(E|I)\) - probability of event given the new information
  • \(P(I|E)\) - probability of the new information given event
  • \(P(I)\) - unconditional probability of the new information
  • \(P(E)\) - prior probability of event

Also, visit our blog for Bayes' theorem explained without formulas (just logical thinking!).

Bayes' Formula - Example

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Example 1 (Bayes' formula)

A car has an alarm that goes off with the probability of 0.8% regardless of whether there is a threat of theft, or not. At a moment when a thief is trying to steal the car, the alarm goes off with a probability of 75%. The probability that the alarm will go off when the thief is not trying to steal the car is 0.050505%. Additionally, we know that the probability that the thief is going to steal the car is 1%.

What is the probability that the thief is trying to steal the car when the alarm has gone off and what is the probability that nobody is trying to steal the car when the alarm has gone off?

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CFA Exam: Bayes' Formula in Finance

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In finance, Bayes' formula is often applied in situations when we have certain knowledge about the probability of an event and, having acquired new information that affects such an event, we update this probability. In such cases the following way of thinking about the formula may be useful:

Updated probability of an event given the new information (posterior probability) is equal to the probability of the new information given event divided by the unconditional probability of the new information times prior probability of an event.

Please note that the posterior probability is equal to the prior probability only when the additional information is in no way relevant to the given event. In other scenarios, the posterior probability is always higher than the prior probability. These observations seem logical. The more useful and reliable information we get, the more accurate our estimation of probability becomes.

Level 1 CFA Exam Takeaways: Bayes' Formula

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  1. Bayes' formula is a way to update probability based on new information.
  2. BAYES' FORMULA: The updated probability of an event given the new information (posterior probability) is equal to the probability of the new information given event divided by the unconditional probability of the new information times prior probability of event.