International Baccalaureate Mathematics

Syllabus Content

Use of Bayes’ theorem for a maximum of three events.

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Probability Probability is a measure of the weight of evidence, and is arrived at through reasoning and inference. In simple terms it is a measure or estimation of likelihood of the occurrence of an event. The word probability comes from the Latin word probabilitas which is a measure of the authority of a witness in a legal case. Some of the earlier mathematical studies of probability were motivated by the desire to be more profitable when gambling. Today however the practical uses of probability theory go far beyond gambling and are used in many aspects of modern life. We believe that even adults can, in many cases, have a poor intuition regarding the effects of probability. These activities are designed to help pupils calculate but also get a 'feel' for the principles of probability.

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Formula Booklet: 4.13

Bayes’ theorem

$$ P(B|A) = \frac{P(B)P(A|B)}{P(B)P(A|B)+P(B')P(A|B')} $$ $$ P(B_i|A) = \frac{P(B_i)P(A|B_i)}{P(B_1)P(A|B_1)+P(B_2)P(A|B_2)+P(B_3)P(A|B_3)} $$

Bayes' theorem provides a way to determine the probability of an event occurring based on prior knowledge of conditions that might be related to the event. When considering three events, Bayes' theorem can be applied iteratively to update our beliefs about the probabilities of each event as new evidence is presented.

The general formula for Bayes' theorem is:

$$ P(A|B) = \frac{P(B|A) \times P(A)}{P(B)} $$

Where:

$ P(A|B) $ is the probability of event A given that event B has occurred.

$ P(B|A) $ is the likelihood of event B occurring given that event A has occurred.

$ P(A) $ is the probability of event A.

$ P(B) $ is the probability of event B.

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International Baccalaureate Mathematics

Statistics and Probability

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