# Contents

- Contents
- Illustration
- Conditional Probability
- Probability of A And B
- Probability of A Or B
- Conditional Probability Revisited I
- Conditional Probability Revisited II
- Independent Events
- Mutual Exclusive Events

# Illustration

# Conditional Probability

## Probability of B Given A

\[\P(B\Vert A)\]- 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

\[\P(A\Vert B)\]- If $B$ occured, what is the probability that now also $A$ occurs?
- Percentage of $B$ that is also in $A$.

# Probability of A And B

\[\begin{align*} \P(A \cap B) & = \P(A) \cdot \P(B\Vert A) \\ & = \P(B) \cdot \P(A\Vert B) \end{align*}\]# Probability of A Or B

\[\P(A \cup B) = \P(A) + \P(B) - \P(A \cap B)\]# Conditional Probability Revisited I

## Probability of B Given A (I)

\[\P(B\Vert A) = \frac{\P(A \cap B)}{\P(A)}\]Deduced from this formula for $\P(A \cap B)$.

## Probability of A Given B (I)

\[\P(A\Vert B) = \frac{\P(A \cap B)}{\P(B)}\]Deduced from this formula for $\P(A \cap B)$.

# Conditional Probability Revisited II

## Probability of B Given A (II)

\[\P(B\Vert A) = \frac{\P(B) \cdot \P(A\Vert B)}{\P(A)}\]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)

\[\P(A\Vert B) = \frac{\P(A) \cdot \P(B\Vert A)}{\P(B)}\]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).