Editing Probability (section)

===Conditional probability===
''[[Conditional probability]]'' is the probability of some event ''A'', given the occurrence of some other event ''B''. Conditional probability is written <math>P(A \mid B)</math>, and is read "the probability of ''A'', given ''B''". It is defined by<ref>Olofsson (2005) p. 29.</ref>

<math display="block">P(A \mid B) = \frac{P(A \cap B)}{P(B)}\,</math>

If <math>P(B)=0</math> then <math>P(A \mid B)</math> is formally [[undefined (mathematics)|undefined]] by this expression. In this case <math>A</math> and <math>B</math> are independent, since <math>P(A \cap B) = P(A)P(B) = 0.</math> However, it is possible to define a conditional probability for some zero-probability events, for example by using a [[σ-algebra]] of such events (such as those arising from a [[continuous random variable]]).<ref>{{Cite web |title=Conditional probability with respect to a sigma-algebra |url=https://www.statlect.com/fundamentals-of-probability/conditional-probability-as-a-random-variable |access-date=2022-07-04 |website=statlect.com}}</ref>

For example, in a bag of 2 red balls and 2 blue balls (4 balls in total), the probability of taking a red ball is <math>1/2;</math> however, when taking a second ball, the probability of it being either a red ball or a blue ball depends on the ball previously taken. For example, if a red ball was taken, then the probability of picking a red ball again would be <math>1/3,</math> since only 1 red and 2 blue balls would have been remaining. And if a blue ball was taken previously, the probability of taking a red ball will be <math>2/3.</math>