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Oct 17

## Reevaluating probabilities based on piece of evidence

This actually has to do with the Bayes' theorem. However, in simple problems one can use a dead simple approach: just find probabilities of all elementary events. This post builds upon the post on Significance level and power of test, including the notation. Be sure to review that post.

Here is an example from the guide for Quantitative Finance by A. Patton (University of London course code FN3142).

Activity 7.2 Consider a test that has a Type I error rate of 5%, and power of 50%.

Suppose that, before running the test, the researcher thinks that both the null and the alternative are equally likely.

1. If the test indicates a rejection of the null hypothesis, what is the probability that the null is false?

2. If the test indicates a failure to reject the null hypothesis, what is the probability that the null is true?

Denote events R = {Reject null}, A = {fAil to reject null}; T = {null is True}; F = {null is False}. Then we are given:

(1) $P(F)=0.5;\ P(T)=0.5;$

(2) $P(R|T)=\frac{P(R\cap T)}{P(T)}=0.05;\ P(R|F)=\frac{P(R\cap F)}{P(F)}=0.5;$

(1) and (2) show that we can find $P(R\cap T)$ and $P(R\cap F)$ and therefore also $P(A\cap T)$ and $P(A\cap F).$ Once we know probabilities of elementary events, we can find everything about everything.

Figure 1. Elementary events

Answering the first question: just plug probabilities in $P(F|R)=\frac{P(R\cap F)}{P(R)}=\frac{P(R\cap F)}{P(R\cap T)+P(A\cap T)}.$

Answering the second question: just plug probabilities in $P(T|A)=\frac{P(A\cap T)}{P(A)}=\frac{P(A\cap T)}{P(A\cap T)+P(A\cap F)}.$

Patton uses the Bayes' theorem and the law of total probability. The solution suggested above uses only additivity of probability.