Bayes' theorem is a mathematical concept that describes the relationship between the probability of an event and the probability of the causes of that event. It provides a method for updating the probabilities of hypotheses in light of new evidence. Bayes' theorem is named after Thomas Bayes, an 18th-century statistician and mathematician.

The formula for Bayes' theorem is as follows:

P(A | B) = P(B | A) * P(A) / P(B)

where:

- P(A | B) is the posterior probability, or the probability of event A given that event B has occurred
- P(B | A) is the likelihood, or the probability of event B given that event A has occurred
- P(A) is the prior probability, or the probability of event A before any new evidence is taken into account
- P(B) is the marginal likelihood, or the total probability of event B across all possible values of A

Bayes' theorem can be used in decision making by considering the probabilities of different hypotheses given new evidence and then choosing the hypothesis with the highest posterior probability. For example, in medical diagnosis, Bayes' theorem can be used to calculate the probability of a patient having a certain disease given their symptoms and test results. The results of the calculation can then be used to guide the decision making process and inform treatment options.

In conclusion, Bayes' theorem is a useful tool for updating probabilities in light of new evidence and can be applied in many fields, including medicine, finance, and engineering. It provides a framework for decision making based on probabilistic reasoning, which can be especially useful in situations where uncertainty is present.