How is the conditional probability defined?
What does it mean when a conditional probability is equal to one?
What does the term mutually exclusive mean?
The conditional probability that the random variable $X=x$ given that the random variable $Y=y$ is given by:
<aside> 💡 $P(X=x|Y=y) = \frac{P(X=x\wedge Y=y)}{P(Y=y)}$
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where the numerator in the expression is the probability that $X=x$ and $Y=y$.
Conditional probability can be used to tell if the outcome of one experiment affects the result of a second (different) experiment as is explained in the following video:
https://www.youtube.com/watch?v=MSj5LNtPVYk
All conditional probabilities must be between 0 and 1. Furthermore, if:
$$ P(X=x|Y=y) = 1 $$
then the random variable $X$ is equal to $x$ whenever the random variable $Y$ is equal to $y$. By contrast, if:
$$ P(X=x|Y=y) = 0 $$
then whenever $Y=y$ $X$ will definitely not be equal to $x$ and the events $X=x$ and $Y=y$ are said to be mutually exclusive.
A useful visual way of understanding what conditional probabilities mean involves using Venn diagrams as discussed in the video below:
https://www.youtube.com/watch?v=aVXR6roXBbY&t=1s
<aside> 📌 SUMMARY: Conditional probability is used to quantify how the result from one experiment affects the result from a second different experiment. If the conditional probability of A given B is equal to 0 then the events A and B are said to mutually exclusive.
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