1 Randomness and Probability

Probability comes up in a wide variety of situations. Consider just a few examples.

The probability that you roll doubles in a turn of a board game.
The probability you win the next Powerball lottery if you purchase a single ticket, 4-8-15-16-42, plus the Powerball number, 23.
The probability that a “randomly selected” Cal Poly student is a California resident.
The probability that the high temperature in San Luis Obispo tomorrow is above 70 degrees F.
The probability that the Los Angeles Lakers win the next NBA championship.
The probability that extraterrestrial life currently exists somewhere in the universe.
The probability that you ate an apple on April 17, 2009.

Example 1.1 How are the situations above similar, and how are they different? What is one feature that all of the situations have in common? Is the interpretation of “probability” the same in all situations? The goal here is to just think about these questions, and not to compute any probabilities (or to even think about how you would).

A phenomenon is random if there are multiple potential outcomes, and there is uncertainty about which outcome will occur.
Uncertainty is understood in broad terms, and in particular does not only concern future occurrences.
Many phenomena involve physical randomness, like flipping a coin or drawing powerballs at random from a bin, or in statistical applications of random sampling or random assignment.
But in many other situations, randomness just vaguely reflects uncertainty.
Random does not mean haphazard. In a random phenomenon, while individual outcomes are uncertain, there is a regular distribution of outcomes over a large number of (hypothetical) repetitions.
Also, random does not necessarily mean equally likely. In a random phenomenon, certain outcomes or events might be more or less likely than others.
The probability of an event associated with a random phenomenon is a number in the interval \([0, 1]\) measuring the event’s likelihood or degree of uncertainty. A probability can take any value in the continuous scale from 0% to 100%.
There are two main interpretations of probability.
- Long run relative frequency. The probability of an event can be interpreted as the proportion of times that the event would occur in a very large number of hypothetical repetitions of the random phenomenon.
- Subjective probability. There are many situations where the outcome is uncertain, but it does not make sense to consider the situation as repeatable. In such situations, a subjective (a.k.a., personal) probability describes the degree of likelihood a given individual ascribes to a certain event. Think of subjective probabilities as measuring relative degrees of likelihood rather than long run relative frequencies.
Fortunately, the mathematics of probability work the same way regardless of the interpretation. In either case, the same basic logical consistency requirements must be satisfied.
A simulation involves an artificial recreation of the random phenomenon, usually using a computer. The probability of an event can be approximated by simulating the random phenomenon a large number of times and determining the proportion of simulated repetitions on which the event occurred out of the total number of repetitions in the simulation.

Example 1.2 One of the oldest documented problems in probability is the following: If three fair six-sided dice are rolled, what is more likely: a sum of 9 or a sum of 10?

Explain how you could conduct a simulation to investigate this question.
In 1 million repetitions of a simulation, a sum of 9 occurred in 115392 repetitions and a sum of 10 occurred in 125026 repetitions. Use the simulation results to approximate the probability that the sum is 9; repeat for a sum of 10.
It can be shown that the theoretical probability that the sum is 9 is 25/216 = 0.116. Write a clearly worded sentence interpreting this probability as a long run relative frequency.
It can be shown that the theoretical probability that the sum is 10 is 27/216 = 0.125. How many times more likely is a sum of 10 than a sum of 9?

Example 1.3 As of Jan 4, FiveThirtyEight listed the following probabilities for who will win the 2023 NBA Championship.

Team	Probability
Boston Celtics	19%
Memphis Grizzlies	16%
Philadelphia 76ers	11%
Denver Nuggets	8%
New Jersey Nets	8%
Other

According to FiveThirtyEight (as of Jan 4):

Are the above percentages relative frequencies or subjective probabilities? Why?
What must be the probability that a team other than the above five teams wins the championship? That is, what value goes in the “Other” row in the table?
The Celtics are how many times more likely than the 76ers to win?
What must be the probability that the Celtics do not win the championship? How many times more likely are the Celtics to not win than to win (this ratio is the “odds against” the Celtics winning).
How could you construct a circular spinner (like from a kids game) to simulate the NBA champion according to these probabilities? According to this model, what would you expect the results of 10000 repetitions of a simulation of the champion to look like?

Example 1.4 Suppose your subjective probabilities for the 2023 NBA champion satisfy the following conditions.

The Nuggets and 76ers are equally likely to win
The Grizzlies are 1.5 times more likely than the Nuggets to win
The Celtics are 2 times more likely than the Grizzlies to win
The winner is as likely to be among these four teams — Celtics, Grizzlies, 76ers, Nuggets — as not.

Construct a table of your subjective probabilities like the one in Example 1.3.

The previous examples illustrate two interpretations of probability: long run relative frequencies and subjective probabilities.
We will use these interpretations interchangeably.
With subjective probabilities it is often helpful to consider what might happen in a simulation.
It is also useful to consider long run relative frequencies in terms of relative degrees of likelihood.
Fortunately, the mathematics of probability work the same way regardless of the interpretation.