L04 - Formal Notation For Conditional Probability

Formal Notation For Conditional Probability

Basic Definition

Conditional Probability

$P(A|B)=\boxed{probability\; of\; A\; given\; B}$

• $P(A|B)$ is defined as the probability of A given B.
  • This means that if we know B has already occurred, what is the probability of A?

Formula

Conditional Probability in Formal (Set) Notation

1. Probability of A given B

$P(A|B)=\frac{n(A\cap B)}{n(B)}$ $=\frac{P(A\cap B)}{P(B)}$

2. Probability of B given A

$P(B|A)=\frac{n(A\cap B)}{n(A)}$ $=\frac{P(A\cap B)}{P(A)}$

Examples

Example 1

20 people, 15 are wearing jackets, 10 are wearing hats.

1. How many are wearing both?

• 15 (J) + 10 (H) = 25
• 25 - 20 = 5 ← $J\cap H$

2. Find $P(H|J)$.

• = $\frac{5}{15}$
• = $\frac{1}{3}$

3. Find $P(J|H)$.

• = $\frac{5}{10}$
• = $\frac{1}{2}$

Example 2

This example has been voided as it is just practice of the formula using a two-way table. Please refer to OneNote for the question.

Example 3

In a group of 23 movie goers, 13 bought popcorn, 15 bought cola, and 9 bought both.

1. $P(P|C)$

• $\frac{9}{15}$

• $\frac{3}{5}$

2. $P(C|P)$

• $\frac{9}{13}$

3. $P(C|M)$ ($=P(C)$)

• $\frac{15}{23}$

Two-way tables and venn diagrams are not required for this example, but can be very useful in terms of reasoning and proving the question.