### Chapter 3. Probability: Probability

### Bayes' Theorem

In probability theory and statistics, *Bayes’ Theorem* describes the probability of an event, based on prior knowledge of conditions that might be related to the event.

For example, you might be interested in finding out a patient’s probability of having a liver disease if they are an alcoholic, or the probability of rolling two dice and getting a total score of 7 when the first die rolls a 3.

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Bayes' Theorem

**Bayes' Theorem**:

#\mathbb{P}(A|B) = \cfrac{\mathbb{P}(B|A)\cdot \mathbb{P}(A)}{\mathbb{P}(B)}#

There is a #40\%# chance it will rain on Sunday. If it rains on Sunday, there is a #10\%# chance it will rain on Monday. If it does not rain on Sunday, there is an #80\%# chance it will rain on Monday.

Events #A# and #B# are defined as follows:

- #A =# 'it rains on Sunday'
- #B =# 'it rains on Monday'

b) What is the probability that it rains on Sunday, given that it rains on Monday #\mathbb{P}(A|B)#?

a) #\mathbb{P}(B) = 0.52#

b) #\mathbb{P}(A|B) =0.0769#

The following probabilities are given:

#\mathbb{P}(A)=0.40#

#\mathbb{P}(A^c) = 1 - \mathbb{P}(A) = 0.60#

#\mathbb{P}(B|A) = 0.10#

#\mathbb{P}(B|A^c) = 0.80#

a) The probability that it rains on Monday is the probability that it rains on Monday, given it does rain on Sunday, plus the probability that it rains on Monday, given that is does not rain on Sunday:

\[\mathbb{P}(B) = \mathbb{P}(B|A) \cdot \mathbb{P}(A) + \mathbb{P}(B|A^c) \cdot \mathbb{P}(A^c) = 0.10 \cdot 0.40 + 0.80 \cdot 0.60 = 0.52\]

So there is a #52\%# chance of it raining on Monday, regardless of whether it rains on Sunday or not.

b) To calculate the probability that it rains on Sunday, given that it rains on Monday, make use of Bayes' theorem:

\[\mathbb{P}(A|B) = \cfrac{\mathbb{P}(B|A)\cdot \mathbb{P}(A)}{\mathbb{P}(B)}=\cfrac{0.10 \cdot 0.40}{0.52} = 0.0769\]

In other words, if it rains on Monday, there is a #7.69\%# chance that it rains on Sunday.

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