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# PHIL 102 Probabilty PHIL 102

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This 3 page Class Notes was uploaded by Kevin Thayyil on Sunday April 17, 2016. The Class Notes belongs to PHIL 102 at University of Illinois at Urbana-Champaign taught by David R. Gilbert in Spring 2016. Since its upload, it has received 29 views. For similar materials see Logic and Reasoning in PHIL-Philosophy at University of Illinois at Urbana-Champaign.

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Date Created: 04/17/16

Lecture 18 Probability Wednesday, April 6, 2016 10:05 AM Three kinds of Probability Theory In a nutshell Example Classical theory Probability statementsthat do not require any The probability of picking a experience in the real world and can be heart from a full and fair deck computed purely mathematically. of cards is 1/4 Relative frequency Calculations will depend on actual observations The probability that a 20 year theory of the frequency of the events in question. old man will live to the age 75 is .63 Subjectivist Theory Through years of experience and observation There is a high probability that and also guesswork - there can't be perfect Mary and peter will get information married In each of these there is a different notion of probability being invoked. A priori Classical Theory The probability of picking a heart from a full and fair deck of cards is 1/4 - Probability statementsthat do not require any experience in the real world and can be computed purely mathematically. The fact that experience isn't required is the reason that the theory that results from this kind of understanding of probability is called the a priori theory of probability, or the classical theory of probability In the classical theory everything has a probability that is a fraction from 0 to 1. The numeratorare number of favorable outcomesand the denominatoris the number of possible outcomes. Relative frequency theory The probability that a 20 year old man will live to the age 75 is .63 - Can we know a priori all the possible outcomes? It seemsto definitely not be the case that all possible outcomesare equally likely So how can we go about making probably calculations in these cases? The RFT is made for this kind of occurrences. Instead of relying upon a priori possibilities, our calculations will depend on actual observationsof the frequency of the events in question. On this picture, the probability of an event A is given by the formula P(a) = f/ tn Where f is the number of observed positive outcomesand tn is the total number of observations Example: Using census data over 55 years, and pick 10,000men, then learn that 6300 are still alive today, then our probability will be 6300/10000= 0.63 Ex. Irregular Die with 19 sides, probability of one side: roll it 1000 times The Subjectivist Theory Sometimesneither worked. Ex: Sometimesneither worked. Ex: There is a high probability that Mary and peter will get married. [you can't have multiple trials] Through years of experience and observationand also guesswork - there can't be perfect information Ex: Sports Gambling Horseraces, somehorse has odds of 3 to 1. This means that the racetrack has decided that the probability of that horse winning is 1/4 (3 times loss for every win) The Probability Calculus Finding probability of compound events. Negations For some event h, the probability that h will not occur is 1 minus the probability that it will occur. 1-Pr(h) Ex. The probability of not drawing an ace at random from a deck of cards is: Pr(!A) = 1 -Pr(A) = 1- 1/13 = 12/13 Independent Two events are said to be independent when the outcomesof one doesn’t affect the outcomeof the other. Conjunction with Independence Given the independent events h1 and h2, the probability of both occurring is the product of their individual probabilities: Pr(h1&h2)= Pr(h1) x Pr (h2) The probability of drawing 2 aces in a row (with replacement)is: 1/13* 1/13= 1/169 Conditional Probability The probability that some vent will occur given that h1 has already occurred General Rule of Conjunction Given 2 events the probability of their both occurring, is the probability of the first occurring times the Probability of the second occurring given that the first occurred. Pr(h1 &h2) = Pr(h1) X Pr (h2 | h1) Mutually Exclusive 2 evets are mutually exclusive when it is impossible for them both to occur Ex: Picking an Ace and a King from a deck or rolling a six or a four. Restricted Disjunction Rule The Probability that at least one of two mutually exclusive events occur is the sum of their individual probabilities Pr(h1 V h2) = Pr(h1) + Pr(h2) But not all disjunctions are like this. Consider the following example: Ex: Suppose that exactly half of the class is male and half if female. Assume further that exactly half the class is older than 19, and that this holds steady for males and females. What is the probabilities that a randomly selected student will be a female or over 19? Its is not 1/2 + 1/2 = 1 We are counting females over 19 twice.

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