Introduction to Probability

Learning Outcomes

Describe disjoint events
Describe a conditional probability
Define independent events
Law of Total Probability
Baye’s Theorem

Disjoint Events

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

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 \(\Pr(A|B)\), the probability of A given B has occurred.

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

Example 1: Drawing Cards

We have a standard deck of 52 cards.
Let’s find the probability of drawing an ace, given that the card is a spade.

Example 2: Rolling Dice

Two fair dice are rolled.
Find the probability that the sum is 10, given that the first die shows a 6.

Example 3: Medical Test

A certain disease affects 1% of the population.
A test correctly identifies the disease 95% of the time, but has a 5% false-positive rate.
Find the probability that someone has the disease given a positive test result.

Example 4: Eye Glasses

	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

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

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

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

Diagrams

Example 1: Weather and Rain Forecast

Suppose the weather forecast is either Sunny (S) or Cloudy (C).

\(P(S) = 0.7\), \(P(C) = 0.3\)
Probability of rain given sunny: \(P(R \mid S) = 0.1\)
Probability of rain given cloudy: \(P(R \mid C) = 0.5\)

What is the probability of rain?

Example 2: Disease

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)?

Independent Events

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

Example 1: Eye Glasses

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

Example 2: Drawing with Replacement

A jar contains 5 red balls and 3 blue balls (8 total).
We draw two balls with replacement.

Example 3: Drawing without Replacement

A jar contains 5 red balls and 3 blue balls (8 total).
We draw two balls with replacement.

Baye’s Theorem

Baye’s theorem computes the probability of an event \(B_i\) given event A

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

Example 1: Factory Machines

A factory has three machines producing items:

Machine A produces 40% of items, with a defect rate of 2%.
Machine B produces 35% of items, with a defect rate of 3%.
Machine C produces 25% of items, with a defect rate of 5%.

If an item is defective, what is the probability it came from Machine C?

Example 2: Email Spam Filtering

Suppose 80% of emails are legitimate, 20% are spam. A keyword appears in 60% of spam emails but in only 5% of legitimate emails. If an email contains the keyword, what is the probability it is spam?

Example 3: Disease

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?