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 90 degrees F.
- The probability that Hurricane Hermine makes landfall in the U.S. in 2022.
- The probability that the San Francisco 49ers win the next Superbowl.
- The probability that President Biden wins the 2024 U.S. Presidential Election.
- The probability that extraterrestrial life currently exists somewhere in the universe.
- The probability that Alexander Hamilton actually wrote 51 of the Federalist Papers. (The papers were published under a common pseudonym and authorship of some of the papers is disputed.)
- 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 Jun 17, FiveThirtyEight listed the following probabilities for who will win the 2022 World Series.
| Team | Probability |
|---|---|
| Los Angeles Dodgers | 20% |
| New York Yankees | 20% |
| Houston Astros | 9% |
| San Diego Padres | 8% |
| Other |
According to FiveThirtyEight (as of June 17):
- Are the above percentages relative frequencies or subjective probabilities?
- According to this model, what would you expect the results of 10000 repetitions of a simulation of the World Series champion to look like?
- The Dodgers are how many times more likely than the Padres to win?
- What must be the probability that the Dogders do not win the World Series? How many times more likely are the Dodgers to not win than to win (this ratio is the odds against the Dodgers winning).
- What must be the probability that one of the above four teams wins the World Series?
- What must be the probability that a team other than the above four teams wins the World Series? That is, what value goes in the “Other” row in the table?
Example 1.4 Suppose your subjective probabilities for the 2022 World Series champion satisfy the following conditions.
- The White Sox and Brewers are equally likely to win
- The Astros are 1.5 times more likely than the White Sox to win
- The Dodgers are 2 times more likely than the Astros to win
- The winner is as likely to be among these four teams — Dodgers, Astros, White Sox, Brewers — 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.