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Appendix

1. What do probabilities mean? The chance of snow tomorrow is 60%. * On past occasions, when circumstances were as they are today, it snowed the next day on 60% of occasions. With either outcome (snow or no snow), one adjusts the model.

2. Outcome: The result of an experiment (e.g., tossing a coin twice, shuffling a deck of cards). Event: A collection of outcomes (where a collection can be just a single outcome).

Events (for two coin tosses): Two heads, HH One head, HT, TH At least one head, HH, HT, and TH

3. How many outcomes are there when one shuffles a deck of cards? A, 2 2, A (2 x 1 = 2)

A, 2, 3 A, 3, 2 2, A, 3 2, 3, A 3, A, 2 3, 2, A (3 x 2 = 6)

A, 2, 3, 4 A, 2, 4, 3 A, 3, 2, 4 A, 3, 4, 2 A, 4, 2, 3 A, 4, 3, 2… (4 x 6 = 24)

The possible arrangement of n cards = n x (n-1) x (n-2) x … x 2 x 1 = n! A standard deck has 52 cards. The number of possible orderings is 52!

4. Suppose you are in a room with 100 people. You give each person a deck of cards and ask him/her to shuffle the deck. What are the odds that two of the decks will have the cards in exactly the same order? Circle one:

Less than 50% [100%]

Greater than 50% [0%]

Suppose you repeat this experiment with 1000 people. Once again, what are the odds that two of the decks will have the cards in exactly the same order? Circle one:

Less than 50%

Greater than 50%

For Greater than 50% This question alone: 15.8% This question second: 41.7%

5. Principles of probability theory Principle I: 0 ≤ p(A) ≤ 1 Principle IIa: p(S) = 1 (Something must happen.) Principle IIb: p(Ø) = 0 (Nothing cannot happen.)

6. Some events are mutually exclusive Principle III: If {A and B} = Ø (equivalently, p[A and B] = 0), then p(A or B) = p(A) + p(B).

7. Conditional probability, p (A | B) Principle IV: p (A | B) = p (A and B)/ p (B) p (A | B) need not equal p (B | A) Given that Pat is a woman (B), what is the probability that she is wearing blue (A)? Given that Pat is wearing blue (A), what is the probability that she is a woman?

8. Slovic et al. A patient—Mr. James Jones—has been evaluated for discharge from an acute civil mental health facility where he has been treated for the past several weeks. A psychologist whose professional opinion you respect has done a state-of-the-art assessment of Mr. Jones. Among the conclusions reached in the psychologist’s assessment is the following: Of every 100 patients similar to Mr. Jones, 10% are estimated to commit an act of violence to others during the first several months of discharge. [vs.] Of every 100 patients similar to Mr. Jones, 10 are estimated to commit an act of violence to others during the first several months of discharge.

Is Mr. Jones high risk, medium risk, or low risk for “harming someone other than himself during the first several months following discharge?” Indicated “low risk” 10% 10

30.3% 19.4%

9. Fair warning: We (i.e., members of the human species) aren’t good at randomness. a. We don’t know what random patterns look like. When I flip a coin, how likely is it to come up heads? HHHHH HTHHT Which event is more probable when a coin is flipped five times? Five heads Three heads and two tails The gambler’s fallacy

10. We don’t want life to be random (i.e., we believe that the world has many more casual relationships than it does). Some eerie coincidences: How would one calculate the prior odds of such coincidences?

11. How many people must there be in a room for the probability to be higher than 50% that two of them will share the same birthday?

With

people in the room, there are

possible pairs. The number of pairs = [n * (n-1)]/2

How many people must there be in a room for the probability to be higher than 50% that one of them will share your same birthday?

12. Teigen & Keren (2020): Are random events perceived as rare? On the relationship between perceived randomness and outcome probability What we perceive as a coincidence depends on the perspective from which the events are told.

Version A: A student named Oscar is travelling from Oslo to London. He is a great fan of Jo Nesbø, a famous Norwegian writer of crime novels, and is reading his latest book. He discovers that the guy seated next to him in the plane is actually Jo Nesbø himself. Now he can ask him to sign the book. Version B: Jo Nesbø, a famous Norwegian writer of crime novels, is traveling from Oslo to London. He discovers that the guy seated next to him in the plane, a student named Oscar, is actually reading his latest book. Oscar is a great fan of Nesbø and can now ask him to sign the book. How likely is this outcome?: 1 (extremely unlikely) to 7 (extremely likely). How random is this outcome?: 1 (not at all random) — 7 (totally random).

When you contemplate a coincidence is there a way to shift the perspective so that it seems less random?...