Math in Politics: Background You’ll Need 2

  • Apply basic probability concepts

Probability

events and outcomes

  • The result of an experiment is called an outcome.
  • An event is any particular outcome or group of outcomes.
  • A simple event is an event that cannot be broken down further.
  • A compound event is a combination of two or more simple events.
  • The sample space is the set of all possible simple events.
If we roll a standard [latex]6[/latex]-sided die, describe the sample space and some simple events.

basic probability

Given that all outcomes are equally likely, we can compute the probability of an event [latex]E[/latex] using this formula:

[latex]P(E)=\frac{\text{Number of outcomes corresponding to the event E}}{\text{Total number of equally-likely outcomes}}[/latex]

 

Probabilities can be expressed as decimals, fractions, or percentages.

Notation: The probability of an event is notated as [latex]P(\text{event})[/latex]

Probability, likelihood, and chance are related concepts that are often used interchangeably, but they have distinct meanings. Probability is a mathematical concept used to quantify the likelihood of an event occurring, likelihood is a measure of how well a hypothesis or model fits the data, and chance is an informal way of expressing the uncertainty of an event. For this lesson, we will be focusing on probability.

certain and impossible events

  • An impossible event has a probability of [latex]0[/latex].
  • A certain event has a probability of [latex]1[/latex].
  • The probability of any event must be [latex]0\le P(E)\le 1[/latex]

Probability of Two Independent Events

Suppose we flipped a coin and rolled a die, and wanted to know the probability of getting a head on the coin and a [latex]6[/latex] on the die. Take a moment to think about how you could approach finding the probability. You could start by listing all the possible outcomes: [latex]{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}[/latex] and determining the total number of outcomes. Notice, out of the twelve outcomes only one is the desired outcome, so the probability is [latex]\frac{1}{12}[/latex]. This example contains two independent events since getting a certain outcome from rolling a die had no influence on the outcome from flipping the coin. When two events are independent, the probability of both occurring is the product of the probabilities of the individual events.

independent events

Events [latex]A[/latex] and [latex]B[/latex] are independent events if the probability of event [latex]B[/latex] occurring is the same whether or not event [latex]A[/latex] occurs.

[latex]P(A \text{ and } B)[/latex] for independent events

If events [latex]A[/latex] and [latex]B[/latex] are independent, then the probability of both [latex]A[/latex] and [latex]B[/latex]  occurring is:

 

A venn diagram with the intersection of sets A and B is shown. The intersection is labeled as A and B

[latex]P\left(A\text{ and }B\right)=P\left(A\right)\cdot{P}\left(B\right)[/latex]

 

where [latex]P(A \text{ and } B)[/latex] is the probability of events [latex]A[/latex] and [latex]B[/latex] both occurring, [latex]P(A)[/latex] is the probability of event [latex]A[/latex] occurring, and [latex]P(B)[/latex] is the probability of event [latex]B[/latex] occurring.

 

Notation: [latex]P\left(A\text{ and }B\right)[/latex] can also be notated as [latex]P(A \cap B)[/latex]

[latex]P(A \text{ or } B)[/latex] for independent events

A venn diagram of the union of sets A and B is shown
The probability of either [latex]A[/latex] or [latex]B[/latex] occurring (or both) is:

 

[latex]P(A\text{ or }B)=P(A)+P(B)–P(A\text{ and }B)[/latex]

 

Notation: [latex]P\left(A\text{ or }B\right)[/latex] can also be notated as [latex]P(A \cup B)[/latex]

Conditional Probability

So far we have computed the probabilities of events that were independent of each other. We saw that getting a certain outcome from rolling a die had no influence on the outcome from flipping a coin, even though we were computing a probability based on doing them at the same time. In this section, we will consider events that are dependent on each other, called conditional probabilities.

conditional probability

The probability the event [latex]B[/latex] occurs, given that event [latex]A[/latex] has happened, is represented as:

 

[latex]P(B | A)[/latex]

 

This is read as “the probability of [latex]B[/latex] given [latex]A[/latex].”

 

Conditional Probability Formula:

If Events [latex]A[/latex] and [latex]B[/latex] are not independent, then [latex]P(A \text{ and } B) = P(A) · P(B | A)[/latex]

It’s important to remember the conditional probability formula can also be written as [latex]P(A \text{ and } B) = P(B) · P(A|B)[/latex].

What is the probability that two cards drawn at random from a deck of playing cards will both be aces?