Last time …
… I wrote of titles. Alex B. in his reply reinforces the religious nature of hockey in Quebec. A puzzle for another time, though I am reminded of what Uli Locher at McGill said in his sociology of religion class – he wasn’t going to define religion except by self-description by participants. I realize in retrospect that this is more challenging than it sounds. Inuit often do not regard themselves as having been religious before Christianity, and he gave a unit on “the American civil religion” (which is certainly not described as a religion by a lot of participants in it!). Maybe next time I will tackle “actor’s categories” again. I am revisiting a topic we have dealt with sometimes, also in light of “natural language” parsing by computers, etc.
Probability (redux?)
Probability is called “the very guide of life” by Joseph Butler, which has been quoted and paraphrased many times since. But what is this about? Three broad notions exist, related to, but not identical, to positions about the foundations of statistics. These are often called “frequentist”, “Bayesian” and “propensity-ist” and are often associated with various figures in the history of thought. They divide into subcategories very quickly. I will not explain all these notions here (or their history), but instead try to encourage creating your own view based on the following. I will use the questions in a future column to analyze the viewpoints that exist in the literature.
These are, therefore, a bunch of (short?) puzzles meant to shake your complacency. There are a lot of “tricks” here, so be warned! I have not included some of the famous puzzles, like Newcomb’s problem or the Monty Hall problem, but some of what they include as challenge is implicitly in some of these.
1. What is the probability of throwing a 6 on a fair die?
2. What is the probability of tossing “heads” with a Canadian 25 cent piece?
3. What is the probability, according to the Royal Canadian Mint, of tossing heads with a Canadian quarter?
4. What is the probability of tossing “tails” with an American quarter?
5. What is the probability of tossing two heads when two quarters are tossed simultaneously?
6. What is the probability of tossing two heads when two quarters are tossed sequentially?
7. What is the probability of tossing half heads, half tails when n quarters are tossed simultaneously?
8. Boy (human) babies are born with what probability?
9. What is the probability that I will bore you with this column?
10. What is the probability of rain tomorrow?
11. An electron is fired at a refracting crystal. What is the probability it will emerge at less than a given angle theta?
12. What’s the probability of god existing?
13. What’s the probability of a thin dart hitting a given area A of a dart board. (Hint: the area of the board is 2A.)
14. What is the probability that all the oxygen molecules in my apartment will be in the upper left outside corner (taken, say, as a cube a meter on a side) in the next minute?
15. What is the probability of Gödel’s version of the ontological argument? (Hint: This one will likely (!) strike the reader as the most odd of all.)
16. What is the probability of you answering this question?
17. Given that an even number was rolled on a fair die, what is the probability that the die used had an odd number of faces?
18. What’s the probability of spinning 1 on a roulette wheel if I just watch the game?
19. What is the probability of spinning 1 on a roulette wheel if I cheer for number 2?
20. The probability of a harmful mutation for the current generation of a population of fruit flies is x. What is the probability that n flies have the same mutation?
21. I will probably go to Quebec in a few days. Calculate this probability.
22. What’s the probability that Vancouver will win the Stanley Cup?
23. What’s the probability that the Cincinnati Reds will win the Stanley Cup?
24. Is the probability in question 23 greater or less than that of the Ottawa Senators winning the World Series?
25. What is the probability that my most-often taken bus (the 11, here in Ottawa) will arrive within 10 minutes? within 30? within 24 hours?
26. A popular commuter train in Tokyo is listed as arriving every 1 minute and 30 seconds. What is the probability it will arrive within 45 seconds?
27. What’s the probability of rolling a total of at least a given number N using the following pattern: Roll 1 six sided die. If it comes up 1-5, roll that many six sided dice and add them up, if they show less than 6. For every 6 in a given context, roll another die and add that to either the total of dice to roll or the total, accordingly. (Bonus for those who know descriptive statistics: What’s the expected value here?)
28. A hunter leaves from a starting point and goes 1 kilometer south. She turns 90 degrees to the right and goes another kilometer west, before turning north again (via a 90 degree turn) and returning to her starting point. Upon returning, she shoots a bear. What is the probability that the bear is brown?
29. “When you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth.” – comment on this classic line.
30. What is the probability that the sun will rise tomorrow?
Certainly, these problems will make us all think long and hard. For the coin tossing questions, given a fair (unbiased) coin and a proven consistent method of flipping the coin, statistical theory says that an infinite number of tosses will give exactly 50% heads and 50% tails. For a fixed number of tosses, the results will vary, but it gets closer and closer to 50-50 as your number of tosses approaches infinity. For most of the other enigmas given here, the probabilities might only be calculable if there exists a body of statistics from previous events. Finally, question #30, the sun will rise tomorrow with 100% probability if you can perceive it with your senses. The sun will not have risen tomorrow if you died in your sleep.