Pascal Continued, and Review
Recall that Pascal is looking for good reasons to believe that God exists. Normally in this class we’ve considered good reasons to be either a priori or a posteriori. Both of these are kinds of evidential reasons. Pascal realizes that there is basically no evidential reason to believe that God exists, but he thinks he can still come up with a good reason to believe. He does this by looking at prudential reasons.
A reason that appeals to what’s in your best interest is a prudential reason. A reason that appeals either to a priori or a posteriori evidence is an evidential reason.
Prudential reasons are offered in support of actions. Pascal says, “Hey, let’s think of believing as a kind of action, and see if there are prudential reasons in favor of it! That would be neat, because then there would be good reasons to believe in God, even without evidence!”
So, Pascal’s argument is distinct among arguments for God in that it grants the atheist that there is no evidence for God’s existence.
Expected Utility (p. 303-304)
The general formula for finding Expected Utility (or Expected Monetary Value) is:
Pr(win) x (net gain) – Pr(lose) x (net loss)
The probability that you win is calculated based on the rules of probability, Ch. 11, and the nature of the bet. So, if you are betting on what two cards you will draw consecutively and without returning the first, then you use Rule 2G, Conjunction in General. If you are betting on which single card you will draw in one shot, then you simply have a probability of winning equal to 1/52. If you are betting that you draw one card or another, then you use Rule 3, Disjunction with Exclusivity.
Calculate the probability that you lose by doing 1 – Pr(win). This is 1, Negation.
The net gain is the payoff of the bet minus the cost of the bet. So, if the payoff is $26, and the bet is $1, then the net gain is $25.
The net loss is usually just the cost of the bet. If you make a $1 bet, and lose, then the net loss is just $1.
Bayes Theorem (p. 291 – 297)
In order to calculate the Pr(h|e), you need three other numbers. Recall that Pr(h|e) is the probability that some belief/hypothesis is true, given some evidence/observation. For example, if you get a positive test result for cancer, then you want to know the probability that you have cancer, h, given a positive test result, e: Pr(h|e).
The three numbers you need to know:
1. Pr(h): The base rate. This is the probability that any random asymptomatic person in your age range has cancer. No test result information is included in this number.
2. Pr(e|h): The sensitivity or reliability of the test. This is a fact about the test. Assuming some person has cancer, how probable is it that the test will deliver a positive result? It is the true positive. You can also calculate this number if you are given the false negative rate. The false negative rate is the Pr(~e|h), the probability that the test comes back negative, given that the person has cancer. 1 – the false negative = the true positive.
3. Pr(e|~h): The false positive rate. Also known as the doozy. This is when the test tells you that you have cancer, but you don’t actually have cancer.
With these three numbers, you can calculate, or better yet estimate with pretty good accuracy, the chance that you have cancer, given a positive test result. This does not apply only to cancer, though. It applies to any sort of similar reasoning. For example, this same Bayes Theorem can help one reason about positive results to a home pregnancy test, a DNA test, and even legal cases.
With the above three numbers, you can use the tree method of estimate the Pr(h|e). Begin with Pr(h). It will usually be a pretty small number, like .007. This number can be expressed by saying “seven thousandths”. This means, 7 out of 1,000. So, let’s imagine that we have a sample size of 1,000 people.
h: 7 ~h: 993
Next, we look at Pr(e|h). It tells us, of the people that have cancer (or whatever), how many will get a positive test result? It is usually a pretty high number, like .9, but it doesn’t have to be. Suppose it is .9. This means the test is 90% reliable. That’s pretty good, but it’s not the whole story, as we’ll see. Of the 7 with the cancer, about 6 will get a positive test result, where 6 is roughly 90% of 7. This means the other person will get a false negative.
+: 6 -: 1
Then, we use the Pr(e|~h), the false positive rate. Of the people who do not have the cancer, how many will get a positive test result anyway? This is usually a pretty small number, like .03. So, of the 993 who do not have cancer, about 27 of them will get a positive test result. The rest, 993 – 27 = 966, of the people without cancer will get a negative result.
+: 6 -: 1 +: 27 -: 966.
Now, we wanted to know the chance we have cancer, given a positive test result. So, of all the people who get positive test results, how many actually have cancer? 6 people out of 27 + 6 = 33, or 6/33, or 2/11. So, with those three numbers that we were given, our chance of having cancer, given a positive test result, is 2/11. So, Pr(h|e) = 2/11. That’s pretty low, so there’s no reason to freak out yet. The thing to do is get a second test done. Two false positives in a row are extremely unlikely, and three in a row are even more unlikely.
Will I need a calculator? NO.
Can I use one anyway? Sure.
Remember all the key concepts from earlier exams, like validity, soundness, IBE, induction, philosophical arguments, informal fallacies, etc., are fair game on this exam.
The exam is structure like this: Part 1: Bayes Theorem, Expected Utility, Pascal’s Wager. 13 questions, multiple choice with some worth 2 points, where you show your work.
Part 2: Cumulative. 20 questions, mostly multiple choice, a few true/false, a couple fill in the blanks, and a couple worth 2 where you show your work.
Overall Grading Scale: