## 1.1 Instances of randomness

Exercise 1.1 Each of the following situations involves a probability. How are the various situations similar, and how are they different? What is one feature that all of the situations have in common? If you were to estimate the probability in question, how might you do it? What are some things to consider? The goal here is not to do any calculations but rather to think about, via these examples, similarities and differences of situations in which probabilities are of interest.

1. The probability that a single flip of a fair coin lands on heads.
2. The probability that in two flips of a fair coin exactly one flip lands on heads.
3. The probability that in 10000 flips of a fair coin exactly 5000 flips land on heads.
4. The probability that in 10000 flips of a fair coin “around” 5000 flips land on heads.
5. The probability you win the next Powerball lottery if you purchase a single ticket, 6-7-16-23-26, plus the Powerball number, 4. (FYI: There are roughly1 300 million possible winning number combinations.)
6. The probability you win the next Powerball lottery if you purchase a single ticket, 1-2-3-4-5, plus the Powerball number, 6.
7. The probability that someone wins the next Powerball lottery. (FYI: especially when the jackpot is large, there are hundreds of millions of tickets sold.)
8. The probability that a “randomly selected” Cal Poly student is from CA.
9. The probability that Hurricane Humberto makes landfall in the U.S.
10. The probability that the Los Angeles Chargers win the next Superbowl.
11. The probability that Donald Trump wins the 2020 U.S. Presidential Election.
12. The probability that extraterrestrial life currently exists somewhere in the universe.
13. The probability that you ate an apple on April 17, 2009.
• The subject of probability concerns random phenomena.
• A phenomenon is random if there are multiple potential outcomes, and there is uncertainty about which outcome will occur. Uncertainty is the feature that all the scenarios have in common.
• Uncertainty does not necessarily mean uncertainty about an occurrence in the future. For example, you either ate or apple or not on April 17, 2009, but you’re not certain about it. But if you tended to eat a lot of apples ten years ago, then you might give a high probability to the event that you ate one on April 17, 2009.
• Many phenomena involve physical randomness2, like flipping a coin or drawing powerballs at random from a bin.
• Statistical applications often involve the planned use of physical randomness
• Random selection involves selecting a sample of individuals at random from a population (e.g. via random digit dialing).
• Random assignment involves assigning individuals at random to groups (e.g. in a randomized experiment).
• However, in many other situations randomness just vaguely reflects uncertainty.
• In any case, 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.
• In two flips of a fair coin we wouldn’t necessarily see one head and one tail. But in 10000 flips of a fair coin, we would expect to see close to 5000 heads and 5000 tails.
• We don’t know who will win the next Superbowl, but we can and should certainly consider some teams as more likely to win than others. We could imagine a large number of hypothetical 2019 seasons; how often would we expect the Eagles to win? The Raiders? (Hopefully a lot for the Eagles; probably not much for the Raiders).
• Also, random does not necessarily mean equally likely. In a random phenomenon, certain outcomes or events might be more or less likely than others.
• It’s much more likely that a randomly selected Cal Poly student is from CA than not.
• Not all NFL teams are equally likely to win the next Superbowl.

1. The exact count is 292,201,338. We will see how to compute this number later.

2. We will refer to as “random” any scenario that involves a reasonable degree of uncertainty. We’re avoiding philosophical questions about what is “true” randomness, like the following. Is a coin flip really random? If all factors that affect the trajectory of the coin were known precisely, then wouldn’t the outcome be determined? Does true randomness only exist in quantum mechanics?