Wiki Page, Math 13 - Perez Sandbox : Module 11 - Introduction to Probability Distributions

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2026-01-17T21:15:10-08:00

In our previous discussions of probability, we focused on determining the probability of one outcome at a time, such as estimating the probability of a head in the toss of a weighted coin.

Now we shift our focus to describing the probabilities of all possible outcomes instead of the probability of just one outcome.

Statisticians describe outcomes as variable values. Each variable value has a probability. All possible variable values together with their probabilities are a probability distribution.

EXAMPLE Probability distribution for a fair coin toss

The variable is coin toss. This is a categorical variable with variable values: Head, Tail.

The probability distribution assigns each variable value (H or T) a probability.

In this case, we used theoretical probabilities to create the probability distribution shown in the table.

Probability distribution for a fair coin toss

Coin Toss Outcome | Head | Tail
Probability | 0.5 | 0.5

EXAMPLE Probability distribution for a weighted coin toss

The variable is coin toss. This is a categorical variable with variable values: Head, Tail. The probability distribution assigns each variable value (H or T) a probability.

In this case, we have to estimate the probability of each outcome (H or T) using relative frequencies based on a large number of coin flips. In the table we use empirical probability estimates to create a probability distribution.

Probability distribution for a weighted coin toss

Coin Toss Outcome | Head | Tail
Probability | 201/1000=0.201 | 799/1000=0.799

EXAMPLE Probability distribution for a blood type

Consider the variable blood type. This is a categorical variable with variable values A, B, AB, or O. Using empirical probability estimates (relative frequencies from large samples of randomly chosen individuals), we can estimate the probability of choosing a person with a given blood type.

Using relative frequencies, the Stanford University's Blood Center (bloodcenter.stanford.edu) gives the probabilities of human blood types in the United States as follows:

Probability distribution
for a blood type

Blood Type | O | A | B | AB
Probability | 0.45 | 0.41 | 0.10 | 0.04

We can use the probability distribution to answer probability questions:

Question: People with blood type O can donate blood to people with any other blood type. For this reason, people with blood type O are called universal donors. What is the probability that a randomly selected person from the United States is a universal donor?

Answer: P(universal donor) = P(blood type O) = 0.45. There is a 45% chance that a randomly selected person in the United States is a universal donor.

EXAMPLE Probability distribution for Boreal owl eggs

Boreal owls are common in Canada and Alaska. They are fairly small, averaging 10 inches in length and weighing from 4 to 6 oz. They often make their nests in woodpecker holes. Using relative frequencies from large field observations, we can estimate the probability of a nest containing a certain number of eggs.

The variable is Boreal owl eggs in a nest. This is a quantitative variable with values 0, 1, 2, 3, 4, 5, or 6 eggs. The probability distribution gives the probability that a nest will have from 0 to 6 eggs.

Probability distribution for Boreal owl eggs

Number of Eggs | 0 | 1 | 2 | 3 | 4 | 5 | 6
Probability | 0.2 | 0.1 | 0.1 | 0.25 | 0.25 | 0.05 | 0.05

Immature Boreal owl (Andy Jones, 2009)

This table is also another example of a probability distribution. Each variable value is assigned a probability.

For example, we can answer the following question from this probability distribution

Question: Which is more likely: (1) To find a Boreal owl nest with 3 eggs, or (2) To find a Boreal owl nest with 4 eggs.

Answer: Both of these events are equally likely. P(3 eggs) = P(4 eggs) = 0.25. There is a 25% chance that if you find a Boreal owl nest, it will have 3 eggs. You are equally likely to find a Boreal owl nest with 4 eggs.

Notice the following important facts about probability distributions:

All outcomes are assigned a probability.

The probabilities are numbers between 0 and 1. This makes sense because each probability is a relative frequency (# times a outcome occurs out of the total number of observations).

The sum of all of the probabilities is 1. This makes sense because we have listed all the outcomes. Each probability is a relative frequency and the relative frequencies will add to 1.

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