What is conditional probability

Answer

Conditional probability measures the chance of one event happening given that another specific event has already occurred. In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event is already known to have occurred.

$P(A|B)=\frac{P(A\cap B)}{P(B)},\text{where }P(B)>0$

This formula shows how the sample space shrinks once B is known. The idea behind conditional probability is that it reduces the sample space to the part of the sample space that involves just the given event B.

What is the difference between conditional probability and joint probability?

Joint probability is the probability that two or more events occur together, denoted as P(A∩B). Conditional probability is the probability of one event given that another has occurred, denoted as P(A|B).

Key differences

1. Joint probability considers the simultaneous occurrence of events without any prior condition, such as both drawing a red card and a heart from a deck.
2. Conditional probability updates the likelihood based on known information, such as the chance of a heart given that a red card was drawn first.
3. Joint probability applies to the full sample space intersection, while conditional probability restricts the sample space to the given event.

In the conditional probability P(A|B) we want to find the probability of A occurring after B has already happened. In the conditional probability the sample space is restricted to just event B before we calculate the probability of A in the restricted sample space. In P(A and B) we want to find the probability of events A and B happening at the same time in the unrestricted sample space.

$P$

Rearranging gives the multiplication rule:

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

$<br>$If events A and B are not independent, then the probability of the intersection of A and B (the probability that both events occur) is defined by P(A and B) = P(A)P(B|A). This shows joint probability as a product of conditional and marginal probabilities.

How does conditional probability relate to independent events?

Conditional probability relates to independent events through the principle that independence means the occurrence of one event does not affect the probability of the other. In the case where events A and B are independent (where event A has no effect on the probability of event B), the conditional probability of event B given event A is simply the probability of event B, that is P(B).

For independent events A and B, the conditional probability P(A|B) equals the unconditional probability P(A), and P(B|A) = P(B).

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