\[\newcommand{\P}{\text{P}} \newcommand{\Vert}{\,\vert\,}\]

Contents

• Contents
• Illustration
• Conditional Probability
  • Probability of B Given A
  • Probability of A Given B
• Probability of A And B
• Probability of A Or B
• Conditional Probability Revisited I
  • Probability of B Given A (I)
  • Probability of A Given B (I)
• Conditional Probability Revisited II
  • Probability of B Given A (II)
  • Probability of A Given B (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

\[\P(A \cap B) = \P(A) \cdot \P(B)\]

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

\[\P(A \cup B) = \P(A) + \P(B)\]

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).