A court is investigating the possible occurrence of an unlikely event T. The reliability of two independent witnesses called Alf and Bob is known to the court: Alf tells the truth with probability \alpha and Bob with probability \beta, and there is no collusion between the two of them. Let A and B be the events that Alf and Bob assert (respectively) that T occurred, and let \tau=P(T). What is the probability that T occurred given that both Alf and Bob declare that T occurred? Suppose \alpha=\beta=9/10 and \tau=1/1000. Return the answer up to the thousands decimal.
Relevant Theorem: Bayes' theorem, named after the Reverend Thomas Bayes, is a fundamental concept in probability theory and statistics that describes the relationship between the conditional probabilities of two events. It is used to update the probability of an event or hypothesis based on new evidence or data.

The theorem is mathematically expressed as:

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

Where:
- P(A|B) is the conditional probability of event A occurring given that event B has occurred (also known as the posterior probability).
- P(B|A) is the conditional probability of event B occurring given that event A has occurred.
- P(A) is the probability of event A occurring (also known as the prior probability).
- P(B) is the probability of event B occurring.

In the context of statistics, Bayes' theorem is often used to update the probability of a hypothesis (A) based on new data (B). The prior probability, P(A), represents our initial belief about the hypothesis before observing the data. The likelihood, P(B|A), quantifies how probable the data is, assuming the hypothesis is true. The marginal probability, P(B), is the overall probability of observing the data, considering all possible hypotheses. Finally, the posterior probability, P(A|B), represents our updated belief about the hypothesis after taking the new data into account.

Bayes' theorem is widely used in various fields, including machine learning, medical diagnosis, finance, and decision-making, to update probabilities based on new evidence and make more informed decisions.
0.075