# Introduction to Probability

## Learning Outcomes

• Describe disjoint events

• Describe a conditional probability

• Define independent events

• Law of Total Probability

• Baye’s Theorem

# Disjoint Events

## Disjoint Events

Two events A and B are considered disjoint if $P(A\cap B)=0$. In general terms, only one event can occur, not both.

# Conditional Probability

## Conditional Probability

Let there be 2 events A and B. Given that B has occurred, what is the probability that A occurs? The conditional probability requires there to be at least 2 events and one event must have occurred. Additionally, the events cannot be disjoint. Conditional probabilities are denoted as $P(A|B)$, the probability of A given B has occurred.

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

## Example

 Uses Eye Glasses Needs Glasses Yes No Yes 44 14 No 2 40
• Find the probability of needing glasses

• Find the probability of not using glasses

• Find the probability of not using glasses and needing glasses

• Find the probability of not using glasses, given they need glasses

# Independent Events

## Independent Events

Events A and B are considered independent if $P(A\cap B)=P(A)P(B)$. In other words, The occurrence of one event will not have an effect on the occurrence of the other event.

## Example

Uses Eye Glasses
Needs Glasses Yes No
Yes 44 14
No 2 40

# Law of Total Probability

## Law of Total Probability

The law of total probability allows you to compute the probability of an event A given that the sets {$B_1$, … , $B_n$} partitions event A. The law of total probability is given as

$P(A)= \sum^n_{i=1}P(A\cap B_i)$

$P(A)=\sum^n_{i=1}P(A|B_i)P(B_i)$

## Example

The probability of an individual having a disease, given that they test positive for the disease, is 0.82. The probability of an individual having a disease, given they tested negative, is 0.14. The probability of testing positive is 0.6. What is the prevalence of a disease (probability of having a disease)?

# Baye’s Theorem

## Baye’s Theorem

Baye’s theorem computes the probability of an event $B_i$ given event A

$P(B_i|A) = \frac{P(A\cap B_i)}{P(A)}=\frac{P(A|B_i)P(B_i)}{\sum^n_{i=1}P(A|B_i)P(B_i)}$

## Example

The probability of an having a disease, given that they test positive for the disease, is 0.82. The probability of and individual having a disease, given they tested negative, is 0.14. The probability of testing positive is 0.6. What is the probability of resulting in a false negative?