Quantitative Reasoning in Philosophy Teaching Probability and Decision Theory in Foundations of Logical Reasoning

Quantitative Literacy in Philosophy [Quantitative literacy] requires logic, data analysis, and probability….It enables individuals to analyze evidence, to read graphs, to understand logical arguments, to detect logical fallacies, to understand evidence, and to evaluate risks. Quantitative literacy means knowing how to reason and how to think. – Gina Kolata, Why Numbers Count: Quantitative Literacy for Tomorrow's America (The College Board, 1997)

Quantitative Literacy in Philosophy Quantitative literacy requires one to understand the nature of mathematics and its role in scientific inquiry and technological progress; to grasp sufficient mathematics to understand important scientific and engineering concepts; and to possess quantitative skills sufficient for responding critically to scientific issues in the media and public life. – F. James Rutherford, physics educator

Quantitative Literacy in Philosophy The heart of quantitative literacy is real world problem solving--the use of mathematics in everyday life, on the job, and as an intelligent citizen. Problem solving must be both mathematically defensible and useful in the real world. – Henry Pollak, applied mathematician

Probability

Objectives: 1.Identify common fallacies in probability reasoning. 2. Apply basic probability rules in practical situations.

Fallacies of Probability It's easy to misuse probabilities. Often, we do this by (mis)applying mental heuristics, which are quick rules for evaluation. While these rules will work much of the time, when they go wrong, they can go very wrong. Heuristic: a general strategy for solving a problem or making a decision.

The Gambler’s Fallacy You flipped a coin three times and it landed heads 4 times in a row. What is the probability it will land heads a 5 th time?

The Gambler’s Fallacy The Fallacy: Why are we tempted to say that the probability is much less than 50%? (1)We confuse the probability of each individual event (each flip of the coin) with the probability of the run of 5 “heads” in a row. (2)We reason that the prior run of 5 influences the chances of the next flip: After this streak of bad/good luck, we must be “due” for a tails.

The Gambler’s Fallacy: Misapplication of the Law of Large Numbers The Law of Large Numbers describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

The Gambler’s Fallacy: Misapplication of the Law of Large Numbers Which sequence of coin tosses is most likely random? T H T H T H T H T H T H T H Or H T T T H T H T T H H T H H

Representativeness Heuristic How would you decide if the following two passages were typed deliberately or randomly? “asdf;lkj sdfoi dfkjfasiu aspfhsd safoi sdfpoiuysdf oishj ops isdy sdiuh sadigsdi usdfoiu sadiu sdutysadliuhasdfnkj sdiusdi uysdiuysdfhi.” “In 1815, M. Charles-Francois-Bienvenu Myriel was Bishop of Digne. He was an old man of about seventy-five years of age; he had occupied the see of Digne since 1806.” The first is representative of random typing on a keyboard. The second is representative of deliberate intention to communicate a specific message.

Representativeness Heuristic The chances of randomly generating gibberish are much higher than randomly generating an intelligent message (i.e. if I was just randomly hitting my keyboard). But, what if the question were slightly different: How would you decide which of the following two passages is more likely to be randomly generated? The chances of generating that particular sequence of gibberish is the same as generating a specific sequence (same number of characters) of intelligent, meaningful text.

Representativeness Heuristic Apply this reasoning to card games: Which of the following is more likely to be dealt? Hand 1: 9 of Hearts, 3 of Clubs, 7 of Diamonds, King of Diamonds, 9 of Diamonds. Hand 2: 10 of Hearts, Jack of Hearts, Queen of Hearts, King of Hearts, Ace of Hearts.

Representativeness Heuristic If you ask most people, they would say that the first hand is much more likely to be dealt than the second. But, the two hands have exactly the same likelihood of being dealt in a fair game. The Fallacy: the first hand is unimpressive, while the second is a very good hand. Our reliance on representativeness blinds us to the true probability: any specific hand is as likely to occur as any other.

The Availability Heuristic Which is more likely? 1. In four pages of a novel (about 2,000 words), how many words would you expect to find that have the form “ ing" (seven-letter words that end with “ing”)? 2. In four pages of a novel (about 2,000 words), how many words would you expect to find that have the form " n -" (seven-letter words that have an “n” before the last letter)?

The Availability Heuristic Thinking of seven-letter words that end in “-ing” is much easier. For example….. Running Texting Playing Writing But it turns out that are there more seven-letter words where the sixth letter is “n”. Why?

The Availability Heuristic Every seven-letter word ending in -ing has “n” as its sixth letter. So, if we can come up with one seven-letter word that (1) has n as the sixth number and (2) does not end in -ing, we know that the second group is larger. Here are a few: asinine, quinine, adenine, guanine.

The Availability Heuristic Which baseball team has a best batting average in the league? Even if you know the batting averages of the top hitters, you might not guess correctly. The players that naturally come to mind are the stars on each team. They are more “available” to our memory, so we are tempted to make our decision by comparing the averages of the team’s top hitters-- without accounting for the effect of taking the team average.

The Availability Heuristic Other examples: Medical Diagnostics / evaluating the results of cancer screening tests Ranking risky behavior

Rules of Probability General Formula: Pr(h) = favorable outcomes / total outcomes Example: What is the probability of drawing a 7 from a deck of cards? Here’s what I need to know: Number of cards in a deck = 52 Number of sevens in a deck = 4

Decision Theory

What is Decision Theory? Decision theory is concerned with the reasoning underlying an agent’s choices. Rational response to new evidence (how ought we to adjust our confidence?) How do we define rational belief? How do we assign value/utility to certain outcomes? How do we rank preferences?

Learning Outcomes 1. Students practice developing a method for problem solving. 1.Identification of the problem 2.Obtaining necessary information 3.Production of possible solutions 4.Evaluation of such solutions 5.Selection of a strategy for performance 2. Students learn to differentiate various types of evidence and types of arguments to support claims (deductive/inductive, inference to the best explanation, argument by analogy, etc.)

Rational Belief 0% % (Certainly false) (Certainly true)

Example: A box contains 7 blue marbles, 4 white marbles, and 10 yellow marbles. What is the probability of drawing a blue? Answer: 7/21 = 1/3

Example: What is the probability of rolling a 4 on a fair die? Answer: 1/6

The Rule of Negation Negation describes the probability that an event will not occur. The negation rule is stated Pr(~h) = 1 – Pr(h) The odds of an event not occurring are 1 minus the odds of the event occurring.

Conjunction with Independence The probability of two independent events (h1 & h2) occurring is the probability of h1 multiplied by h2. Why? Because the outcome of the first draw provides no information about the outcome of the second draw, so the events are independent.

Example 1: What are the odds that I'll roll a double six with a pair of dice? Pr(h1 & h2) = Pr(h1) × Pr(h2)

Example 2: What are my odds of drawing two aces with replacement? Pr(two aces) = Pr(drawing one ace) × Pr(drawing a second ace) Pr(two aces) = Pr(1/13) × Pr(1/13) = 1/169

Conjunction without Independence But, what about the probability of drawing two aces when the first card is discarded from the deck? In this case, the events aren't independent. So, we need to calculate conditional probability: Pr(h1 & h2) = Pr(h1) × Pr(h2|h1) = 4/52 X 3/51 = 1/221

Example 1: Let's say that I draw a King from my deck of cards. What are the odds that the next card I draw will be an Ace? Pr(h1 & h2) = Pr(h1) × Pr(h2|h1)

Disjunction with Exclusivity "Exclusivity" means that the events cannot both occur on the same trial. You can't get both a ten and an eight in a throw of two six-sided dice, for example. Either A or B (when it’s impossible for both events to occur at the same time)

Disjunction with Exclusivity What is the probability that a randomly selected student at BSU is either a junior or a senior? Assume that 23% of students are juniors and 20% of students are seniors. Then, the probability that a student is in one of those two categories is … Pr(A or B) = Pr(A) + Pr(B) = 43%

Disjunction with Exclusivity Pr(h1 or h2) = Pr(h1) + Pr(h2) What are the chances that either an eight or a two comes up in a single throw of two six-sided dice? Pr(rolling 8 or rolling 2) = Pr(rolling 8) + Pr(rolling 2) Pr(rolling 8 or rolling 2) = 5/36 + 1/36 = 1/6 How did I get 5/36? For the total number of events, I know that there are 36 possible results in throwing two six-sided dice (6 × 6). For 5, I consulted the chart on page 284 and counted the number of combinations that would equal 8. Likewise for 1/36.

Disjunction with Exclusivity What are my chances of rolling an even number in one throw of a die? There are 3 different ways to roll an even number (2, 4, or 6) There are 6 possible rolls 3/6 = 1/2

Disjunction Without Exclusivity What if we are dealing with compatible events? The probability that at least one of two events will occur is the sum of the probabilities that each of them will occur, minus the probability that they will both occur. Pr(A or B) = Pr(A) + Pr(B) – Pr(A and B) If A and B are mutually exclusive, then Pr(A and B) = 0

Disjunction (In General) What are the odds that a BSU student is either a junior or a philosophy major? Assume that juniors are 23% of all BSU students. Assume that philosophy majors are 5% of the all BSU students. We can’t just add because we’ll end up double-counting the philosophy majors who are also juniors.

Mutually Exclusive vs. Independent Two events are mutually exclusive if and only if they cannot both occur at the same time. Two events are independent if and only if the occurrence of one does not affect the probability of the other.