Conditional Probability

1. Overview

In probability theory, conditional probability is a measure of the probability of an event occurring given that another event has (by assumption, presumption, assertion or evidence) occurred. The concept of conditional probability is one of the most fundamental and one of the most important in probability theory. But conditional probabilities can be quite slippery and require careful interpretation. For example, there need not be a causal relationship between A and B, and they don't have to occur simultaneously.

2. Description

2.1 Definition

$$P(A\: |\: B)=\frac{P(A\cap B)}{P(B)}$$

The likelihood of an event occuring, assuming a different one has already happened

where $P(B)> 0$ and $P(A\cap B)$ is the probability that both events A and B occur

2.2 Law of total probability

$$P(A\: |\: B)\neq P(B\: |\: A)$$

$$A=B_{1}\cap B_{2}\cap \cdots B_{n}\\
P(A)=P(A\: |\: B_{1})\times P(B_{1})+P(A\: |\: B_{2})\times P(B_{2})\cdots $$

2.3 Additive law

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

The probability of the union of two sets is equal to the sum of the individual probabilities of each event, minus the probability of their intersection.

2.4 Multiplication rule

$$P(A\: |\: B)\times P(B)=P(A\cap B)$$

3. Explanation

3.1 Coin flipping(Indepent event)

3.2 Queen of spades

4. Property

4.1 P(B) = 0

4.2 Order

5. Example

5.1 Conditional probability

5.2 Additive probability

$$P(A\cap B)=P(A)+P(B)-P(A\cup  B)$$

5.3 Multiplication rule

5.3.1 Case 1

5.3.2 Case 2

$$P(A\cap B)=0.255\times 0.75=0.191$$

We have a probability of 0.191 of drawing a spacer on the second turn, assuming we did not draw one initially

6. References

https://en.wikipedia.org/wiki/Conditional_probability#targetText=In%20probability%20theory%2C%20conditional%20probability,%2C%20assertion%20or%20evidence)%20occurred.