# Probability Theory

Daniel Weibel
Created 8 Nov 2017

# Conditional Probability

## Probability of B Given A

• If $A$ occured, what is the probability that now also $B$ occurs?
• Percentage of $A$ that is also in $B$.

## Probability of A Given B

• If $B$ occured, what is the probability that now also $A$ occurs?
• Percentage of $B$ that is also in $A$.

# Conditional Probability Revisited I

## Probability of B Given A (I)

Deduced from this formula for $\P(A \cap B)$.

## Probability of A Given B (I)

Deduced from this formula for $\P(A \cap B)$.

# Conditional Probability Revisited II

## Probability of B Given A (II)

In this formula for $\P(B\Vert A)$, replace $\P(A \cap B)$ with this formula for $\P(A \cap B)$.

This equation is called Bayes’ Theorem.

## Probability of A Given B (II)

In this formula for $\P(A\Vert B)$ replace $\P(A \cap B)$ with this formula for $\P(A \cap B)$.

This equation is called Bayes’ Theorem.

# Independent Events

The happening of one event has no effect on the probability of the other event.

For example:

• $A$ = getting head on first toss of a coin
• $B$ = getting head on second toss of a coin

In this formula for $\P(A \cap B)$, replace $\P(B\Vert A)$ with $\P(B)$.

We can do this, because if $A$ occurred, the probability that $B$ occurs is not affected by that. The probability that $B$ occurs is still $\P(B)$.

# Mutual Exclusive Events

The happening of one event prevents the happening of the other event.

For example:

• $A$ = rolling a 1 with a die
• $B$ = rolling a 2 with a die

In this formula for $\P(A \cup B)$, replace $\P(A \cap B)$ with 0.

We can do this, because events $A$ and $B$ cannot occur together (the probability that they occur together is 0).