Class Notes (838,376)
Canada (510,867)
Philosophy (401)
PHIL 105 (81)
Lecture

An Overview of the Rules of Probability XX1 SP14.pdf

11 Pages
82 Views
Unlock Document

Department
Philosophy
Course
PHIL 105
Professor
Colin Stewart
Semester
Fall

Description
An Overview of Basic Probability Theory: Part I Propositions and Events What sort of thing has a probability? We often talk about the probability of a particular event occurring: “what are the chances that it will snow tomorrow?” But we can also talk about whether or not a proposition has a probability: what is the probability that “it will snow tomorrow” is true? Don’t worry too much for now about distinguishing between propositions and events for the purposes of our class – this is not going to play a huge role in what we do in here. But this difference matters quite a bit as you get into more serious theoretical work in probability theory. Different positions will take different notions as fundamental: some will speak in terms of events (for good reaso n) and some will speak in terms of propositions (also for good reason). We will get into some of this material in the second half of the book – so be prepared for that. But don’t worry too much about it now. Logical Notation We are going to represent pr opositions or events with capital letters: A, B, C…etc. So, for example, we will represent the proposition “it will snow tomorrow” by simply using S. Frequently, we will want to talk about compound events: what is the probability that it is both cold and snowy tomorrow? And so on. Here are the basic ways that we’re going to do that. A v B: A or B will happen (or both) A & B: A and B will happen ~A: A will not happen Probability Notation We will use the notation Pr ( ) to represent the proposition the probability of ___ occurring. Any instance of the logical notation that we’ve just seen can be put between the parentheses. Let’s try an example. Take the proposition “I will receive an A or a B in this class.” We will symbolize that proposition: A v B. Now, if we’re trying to represent “the probability that I will receive an A or a B in this class” we would write: Pr (A v B). Take another example. What is the probability that it will not be cold tomorrow? The proposition is: “it will be cold tomorrow. ” We can symbolize that is C. We’re wondering what the probability of that proposition being false is. So we need to negate the proposition: ~C. And now we need to attach our probability operator to it: Pr (~C).   1   Two Conventions of Probability Theory . 1) All probabilities lie between 0 and 1. The probability of something impossible occurring is 0 and the probability of something certain occurring is 1. It is quite rare indeed to find something that has a probability of 1. 2) The probability of something certain occurring is 1. Mutually Exclusive and Independent Two events are mutually exclusive when they cannot both occur (at the same time). So, for example, the proposition “It is overcast” and the proposition “It is sunny” are mutually exclusive propositions. Here are some more examples: 1) “The wheel will stop on red” and “The wheel will stop on black.” Pr (R & B) = 0 2) “I will receive an A” and “I will receive a B.” Pr(A & B) = 0 Two events are independent when the occurrence of one does not influence the probability of the other occurring. So, for example, the propositions “I will get a phone call today” and “I will eat breakfast today” are independent of one another: it is not plausible to suppose that one occurrence exerts any influence on the other. To put this in terms of some terminology we have already looked at in this course, we can say that event A is independent of event B if A is neither positively relevant nor negatively relevant to the occurrence of B. Adding and Multiplying Probabilities We’re now going to look at the two most basic rules of probability theory – rules that we will formalize in Part III . So these two rules are expressed here in informal terms. For the purposes of assignments and exams, you should appeal to the more formal characterizations of them below. Rule 1: The probabilities of mutually exclusive events add up Take the propositions “I will get an A in this class” and “I will get a B in this class.” Let’s symbolize them as A and B. Let’s suppose that we know that Pr(A) = .1 and Pr(B) = .2. So now you know the probability that you will get an A in this class and you know the probability that you will get a B in this class. But now suppose you want to know what the probability that you will get at least a B in this class is. What is the right way to symbolize this? Here are two options: Pr (A&B) Pr (A v B) Pretty clearly, the first possibility is impossible: you can’t get an A and a B in this course. You get only one grade. So if you’re wondering what the probability is that y ou will get at least a B, then you’re wondering what the probability is that you will get an A or a B: Pr(A v B). And we said above that we can add them up. Pr (A v B) = Pr(A) + Pr(B) . Pr(A) = .1   2   Pr(B) = .2 Pr(A v B) = .3 This rule can be confusing: you are adding probabilities when you are calculating the probability of one or the other of two mutually exclusive events occurring . You will be tempted to apply the addition rule when figuring out Pr(A & B). But just because you’re adding events (in a sense) doesn’t mean you are adding probabilities. So this rule does not apply to Pr(A&B). Rule 2: Multiply probabilities to find the probability of two independent events both occurring Pr (A & B) = Pr(A) x Pr(B) Suppose I know that the probability that I w ill receive a phone call today is .8. And suppose I know that the probability I will eat dinner today is .9. Let’s calculate the probability of both happening. If we add probabilities, we get: Pr(P) + Pr(D) = .8 + .9 = 1.7 . But that means the event is more than certain! This is an impossible probability. Let this be a warning: even though the notation uses “&,” you do not want to add probabilities in this case. The correct way to calculate this is: Pr(P & D) = Pr(P) x Pr(D) = .8 x .9 = .72 . So there is a roughly 3 in 4 chance that I will both eat dinner and receive a phone call today. Beware of dependent events! What is the probability that you will roll both an even and a prime when you roll one die? Pr(E) = 3/6 = 1/2 Pr(P) = 3/6 = 1/2 Pr(E & P) = 1/2 x 1/2 = 1/4 So you have a .25 probability of rolling both an even and a prime with one die? No, this is a mistake: E and P are not independent . They are dependent events. In fact, there is exactly one even prime: 2. So Pr(E&P) = Pr(2) = 1/6. When events are dependent, you may not multiply probabilities – not exactly, anyway. We’ll consider this in more detail below. Practice Question With two dice, how can you roll a 6 or a 7? To roll a 6: (1, 5) (5, 1) (2, 4) (4, 2) (3, 3)   3   To roll a 7: (1, 6) (6, 1) (2, 5) (5, 2) (3, 4) (4, 3) Since there are 36 possible outcomes on a roll of two dice, the probability of rolling a 6 is 5/36 and the probability of rolling a 7 is 6/36 or 1/6. So, with two dice, the probability of rolling a 7 is higher than the p robability of rolling a 6. But most people don’t realize this initially. Practice Question Suppose you have two urns full of marbles. In the first urn, there are 3 red marbles and 1 green marble. In the second urn, you have 1 red marble and 3 green marbl es. So this can be represented as follows: Urn 1: 3 R, 1 G. Urn 2: 1 R, 3 G. Flip a coin to select which urn to draw from and then draw two balls from that urn with replacement. (This means that after you draw the first marble, you note its color and the n put it back into the urn before you draw again.) What is the probability of getting two reds on this setup? We know that the probabilit y of drawing exactly one red ball is 1/2. Pr (R 1 = 1/2 x 3/4 = 3/8. (The probability that you draw from Urn 1 is 1/ 2 – since it depends upon a coin flip – and the probability of then drawing a red marble is 3/4, since there are three red marbles and one green marble.) Pr (R 2 = 1/2 x 1/4 = 1/8 (The probability that you draw from Urn 2 is ½ - since it depends upon a coin flip – and the probability of then drawing a red marble is ¼, since there is one red marble and three green marbles.) Now, applying Rule 1 – which says that we must add the probabilities of mutually exclusive events (and these are mutually exclusive, s ince you cannot draw from both urns) – we get: Pr (R 1 R ) 2 1/8 + 3/8 = 4/8 = 1/2 This is the probability of drawing exactly one red marble. So now we’re wondering about the probability of drawing two reds. Since we know the probability of drawing a single red is 1/2, shouldn’t we just say that the probability of drawing two reds is 1/2 x 1/2 = 1/4? The answer is no. This is a tough question, but the probability is not 1/4. The reason is that there are two different ways of drawing two reds in a row on this setup. First way: Coin flip indicates urn 1; you draw two reds in a row from urn 1 Second way: Coin flip indicates urn 2; you draw two reds in a row from urn 2   4   Pr (First way): 1/2 x 3/4 x 3/4 = 9/32. Here, the probability of drawing from Urn 1 is 1 /2 and then the probability of drawing a red on the first draw is 3/4 . Since we are drawing with replacement, the probability of drawing a red on the second draw is the same. Pr (Second way): 1/2 x 1/4 x 1/4 = 1/32. The same reasoning applies. Now, to figure out the probability of drawing two reds, we must account for both ways of doing so. Since these are mutually exclusive (you cannot draw from both urns), we must add the probabilities together: Pr (F v S) = 9/32 + 1/32 = 10/32 = 5/16 An Overview of Basic Probability Theory: Part II Categorical and Conditional Probability Categorical probabilities state probabilities that contain, we might say, no “ifs, ands or buts.” Here are some examples: The probability it will rain tomorrow is .1 I have a 1/13 chance of drawing a spade Conditional probabilities give the probability of something happening conditional upon something else happening. Here are some examples: The probability that I will be hungover given that I had 12 beers last night The probability that you will walk to school given that it is snowing Notation Categorical Probability: Pr ( ) Conditional Probability: Pr ( / ) The probability that it will rain tomorrow: Pr (R) The probability that I will be hungover given that I had twel ve beers last night: Pr (H/T) So far, we have looked only at categorical probabilities. Now, however, we will look at some additional rules that concern conditional probabilities. Examples Bingo: bingo players calculate conditional probabilities all the time. As they fill up a line on their
More Less

Related notes for PHIL 105

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit