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.

Diagram

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.

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

Diagram

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

Work

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

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

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

Diagrams

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?