NOVEMBER 19, 2013
Imagine you are an adult professional gambler and have the opportunity to play the following game:
“Pay $10 to roll a 5-sided die. You win nothing if you roll a 1, 2, or 3; you get $20 for a 4 and $25 for a 5. Should you play?”
After debating whether a 5-sided fair die could even exist, the few students in the room didn’t even have to complete an expected-value (EV) calculation to determine that the EV would be negative. The answer would be “no.” Another student arrived as I proposed Version 1.1 of the game:
“One side of the die is weighted with lead, so that the possible outcomes are not equally likely. Now you have a 60% chance of getting a 5, while the other outcomes remain equal. Should you play?”
After more debate about how just one side of the apocryphal 5-sided die could come up more often than the others when only one side was weighted, we decided to make the assumption required to answer the question.
The students did not remember how Raissa had shown them to calculate EV for unequal probabilities of risk/gain, so they invented their own method. This method is pretty clear from the accompanying photo, but I’ll try to paraphrase: Since you have 4 outcomes that are each equally 10% likely, and one outcome (a roll of 5) that occurs 60% of the time, we can split the latter single outcome into 6 separate outcomes (still each a roll of 5), each with equal likelihood of 10%. Then we calculate the dollar value of each of our now-ten possibilities and add them up to get positive $70. If you divided $70 by 10 outcomes, you’d get an EV of $7. Pretty brilliant, I must say. To give credit where credit is due, I don’t think I ever would have thought of doing that. One thing I love about Math Circles is that there is no requirement to use one particular method to solve a problem. We have the freedom to do it any way we choose. And math, to quote Bob Kaplan, is freedom.
At this point, the group remembered the method Raissa taught (multiplying the probability of each outcome by the value of each outcome, then adding), and was happy that this result gibed with their own.
A few more students arrived and we finally had enough for a quorum or a minyan or whatever we would call the sufficient number for a Math Circle. (What would you call that?)
I read aloud some excerpts about the life of Blaise Pascal from E.T. Bell’s Men of Mathematics. I prefaced it with a request to keep Bell’s views in historical perspective, as he wrote the book in 1937. Students were shocked at Bell’s provocative language. “Let me reread that aloud,” demanded one student, who then took the book and read this sentence: “Here our point of view must necessarily be somewhat oblique, and we shall consider Pascal primarily as a highly gifted mathematician who let his masochistic proclivities for self-torturing and profitless speculations on the sectarian controversies of his day degrade him to what would now be called a religious neurotic.” (p73) The students were offended, and demanded to know from what facts Bell drew this conclusion. I promised to look into it, and will have more info next time.
My intent, in reading from this book, was to put Pascal’s Wager into historical perspective before exploring the math involved. Fortunately, the rest of the chapter was somewhat more objective, less provocative, and therefore more useful. Students found Pascal’s connection to Descartes and Fermat interesting, but knew nothing about the religious climate in France during the reign of Cardinal Richelieu. Armed with this information, we set about to figure out what was so remarkable about Pascal’s “Wager.”
“God is, or He is not. But to which side shall we incline?...Let us see… Let us weigh the gain against the loss in wagering that God is… If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.”
Since I love Pascal’s language, I read the full quote to the group. Then the kids took over the conversation:
- To use a decision matrix, you have to know the probability of each state/outcome. Let’s attempt to walk in Pascal’s shoes and assume that there’s a 50% chance that God really exists.
- In fact, let’s try to assume that we do live in Pascal’s time and have his beliefs.
- Pascal seems to assume that (1) you can change your beliefs, and that (2) if you could, you would. This assumption is highly suspect.
- A decision matrix requires a numerical value for each outcome. Here I explained a bit about how economists measure “utility” in “utils.”**
- Pascal seems to believe that there is nothing to lose by (1) believing in God if God doesn’t exist or (2) not believing if God does exist. Our group raised many objections to these assumptions.
- What did Pascal mean when he said “All the troubles of man come from his not knowing how to sit still?”
- Why is Pascal’s triangle so famous?
- When was infinity created?
- Can you measure the unmeasurable?* (One student quoted her science teacher, who reminds students that “we can’t measure the unmeasurable.”)
- What is the definition of “belief” in God?
- What is the definition of “God?” and could there be multiple definitions, each with its own probability?
Comments and questions were flying so quickly that I was unable do perform my secretarial duties adequately. We therefore don’t have a good record of questions and assumptions. The group went on to perform two calculations using decision matrices (see photo). The first assumed that 1 “happiness unit” is the reward for believing in an existing God, and that 1 unit was lost for not believing if existence, and for belief in non-existence, resulting in an EV of +1/2. The second calculation assumed infinity happiness units were at stake, still resulting in a positive EV. The group concurred that, had they been Pascal in his time with his beliefs, they would have offered the same advice to “Wager, then, without hesitation.”
*I’ve noticed that many writers seem to almost dismiss Pascal’s wager as a novelty compared to his contribution to the field of probability. He did help to invent probability after all. But in the Wager, isn’t he also asking a Really Big Question that is still (and maybe even more so) relevant in our times: Can we measure the immeasurable? Isn’t this what the behavioral and social sciences try to do?
**This may be my first time using this part of my degree in economics! I love the merger of theoretical and applied math. But I need to work hard here to keep my ego out of facilitating a math circle. I can feel the wisdom Bob and Ellen Kaplan offer about choosing math-circle topics: pick something that you are curious about and know just a little bit about. For the first time, I have to resist interjecting my own knowledge. Boy, that’s hard.
PS For those of you who had children in our Greek Mythology Math Circle, take a look at this video: Math Playtime with Blocks. It takes what we were doing in our exploration of Narcissistic Numbers in another interesting direction.