Conditional Probability

What is Conditional Probability?

In essence, conditional probability is for when an event has already happened, and we want to model what comes next.

Examples:

• It is cloudy and might rain
• Given that you are a felon, probability that you go to college
• Given that you were accepted to college, probability that you receive a scholarship
• Given that you are registered to vote, probability to vote

Similar to a heuristic (you know one thing, how can you predict the next).

Looking into conditional probability is also important to understand the independence of events. It seems backwards, but if you know the probability of something happened based on something else, you can find if there is a correlation or probability increase.

This causes us to ask these two questions:

• Are two events actually related?
• Does knowing one tell us nothing about the other?

The distribution of conditional probability is given by:

Conditional Probability Distributions

The conditional probability $P(A|B)$, which is the probability of event $A$ given that event $B$ has occurred, is given by:

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

Ex: With a six-sided die 🎲

Probability of rolling at least a $3$: $A=\{3,4,5,6\},P(A)=\frac{2}{3}$

Probability of an even roll: $B=\{2,4,6\},P(B)=\frac{1}{2}$

The conditional probability that a roll is at least $3$ given that the roll is even:

$P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{\frac{1}{3}}{\frac{1}{2}}=\frac{2}{3}$

The conditional probability that a roll is even given that it is at least 3 is: $P(B|A)=\frac{P(A\cap B)}{P(B)}=\frac{\frac{1}{3}}{\frac{2}{3}}=\frac{1}{2}$

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