$P(A)$ → prior
$P(B|A)$ → likelihood
$P(A\cap B)$ → probability that both events $A$ and $B$ occur.
$P(A)$ → posterior
Conditional probability is the probability of an event occurring given that another event has already occurred. It's a way of updating our beliefs about the likelihood of an event based on new information.
When $P(A|B)=P(A)$ then $A$ and $B$ are independent.
Note → It’s possible for $A$ and $B$ to be conditionally independent given the occurrence of another event $C$ → $P(A\cap B|C)=P(A|C)P(B|C)$ → that means, given that $C$ has occurred, knowing that $B$ has also occurred tells us nothing about (provides no additional knowledge) the probability of $A$ having occurred → However, conditional independence doesn’t necessarily mean $A$ and $B$ are independent → In other words, conditional independence does not imply unconditional independence → In other words, conditional independence is a concept where $A$ and $B$ are independent of each other given a third event $C$ → the occurrence of $C$ makes $A$ and $B$ independent.
<aside> 💡 INTERVIEW TIP If other information is available and you are asked to calculate a probability, you should always consider using Bayes’ rule.
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Assume we have several disjoint events within $B$ having occurred. Using the law of total probability, we can break down the probability of $A$ as follows:
$$ P(A)=P(A|B_1)P(B_1)+\dots+P(A|B_n)P(B_n)=\sum_{i=1}^n P(A|B_i)P(B_i) $$
<aside> 💡 DEFINITION: DISJOINT EVENTS
In probability, disjoint events, also known as mutually exclusive events, are events that cannot occur at the same time. This means that they have no outcomes in common. For example, the result of a coin toss can be either heads or tails, but not both, so these are disjoint events.
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Random variables are often analyzed with respect to each other → Joint PDFs.
Random variables $X$ and $Y$ varying over a two-dimensional space → the integration of the joint PDF yields the following
$$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f_{X,Y}(x,y)dxdy = 1 $$