Download presentation

Presentation is loading. Please wait.

Published byJosef Burrington Modified over 3 years ago

1
7.4 Basic Concepts of Probability This presentation is copyright by Paul Hendrick © 2003-2005, Paul Hendrick All rights reserved

2
7.4 Basic Concepts of Probability Union rule for probability –Union rule (general case – always true) –IF P(E F) = 0, it can be omitted! –Union rule for mutually exclusive events, only

3
7.4 Basic Concepts of Probability Complement rule –back-door approach (because it doesnt calculate directly)

4
–Probability uses two numbers -- # ways for a successful outcome & # ways for any outcome n(E) and n(S) P(E) = n(E) / n(S) Let S={1,2,3,4,5,6} be the sample space for rolling a single fair die. Let E={1,6} be the event of rolling an extreme number. n(E) = 2; n(S) = 6; so P(E) = 2/6 = 1/3. –There is a third number -- # ways for an unsuccessful outcome n(E) n(E) = n(S) - n(E) For the above, n(E) = 6-2 = 4, the number of ways of NOT rolling an extreme number. Note: P(E) = 4/6 = 2/3 or P(E) = 1 – P(E) = 1-1/3 = 2/3. 7.4 Odds

5
–odds uses a different two of the three numbers –odds in favor of an event E = n(E) / n(E) (assumes all outcomes are equally-likely) Odds in favor of an extreme roll are 2 / 4 or 1/2 = P(E) / P(E) (uniform sample space NOT necessary for this formula) Odds for extreme roll also by 1/3 / 2/3 = 1/2 –odds against an event E = n(E) / n(E) (numerator and denominator switched!) Odds against an extreme roll are 4 / 2 or 2/1 –Again, note the 2 & 4 are reversed from the first example above. 7.4 Odds (cont)

6
–Instead of as fractions, odds are commonly shown as ratios with a colon used to show comparison –n(E) : n(E) instead of n(E) / n(E) –Odds in favor of an extreme roll would then be 2:4 or 1:2 instead of 2/4 or ½ –(read as two to four, or 1 to 2, resp.) –Odds against an extreme roll would then be 4:2 or 2:1 instead of 4/2 or 2/1 or even just 2 –Note in the above, that odds are generally reduced, just as fractions are. 7.4 Odds (cont)

7
–A lot of people confuse odds with probability -- they are similar ideas (and sometimes close numbers), but are not the same. –Recapping the previous example of the event E = the extreme roll of a die, –P(E) = 2/6 = 1/3 ; odds for E are 2:4 or 1:2 –P(E) = 4/6 = 2/3 ; odds against E are 4:2 or 2:1 –Different example, consider F = rolling a sum of 12 on two fair dice: –P(F) = 1/36 ; odds for F are 1:35 –P(F) = 35/36 ; odds against F are 35:1

8
7.4 Odds (cont) –You should understand the similarities and also the differences between odds and probability. –You should be able to calculate both: Odds in favor of an event E (or simply for the event) Odds against an event E –You should be able to convert from probabilities to odds, or vice versa, on a given problem. The book gives some formulas for this on page 349, if you want to do it by formula –Odds are mainly used by gamblers for handling money; were not too concerned with this in class –We will predominantly use probabilities in class – in fact combinations such as union and intersection are much easier to do with probabilities than with odds!

9
7.4 Further probability notions Types of probability –theoretical (by counting in a uniform sample space -- the formula P(E) = n(E) / n(S) ) –empirical (by having observed typical outcomes -- an experiment) In a city study at an intersection, out of the 500 northbound cars, 35 of them turned left. Whats the probability of such a car turning left? The empirical probability = 35 / 500 = 7 / 100 or 7% This kind of probability is sometimes referred to as relative frequency –intuitive ( a gut feeling -- from experience?) You think you have a fifty-fifty chance of acing exam 1. A businessman who has a successful chain of 20 pizza restaurants estimates a new restaurant on Texas at University will have an 85% chance of being successful.

10
Probability distribution for an experiment –Simply a list of all possible outcomes and their associated probabilities –Easiest given as a table –Probability distribution for 1 fair die: –Probability distribution for sum of 2 fair dice: 1/6 4 2 Pr 6531E 1/36 12 5/36 6 1/6 7 5/36 8 1/9 9 1/12 10 1/12 4 1/36 2 1/181/91/18Pr 1153E 7.4 Further probability notions

11
Properties of probability –Let S be a sample space consisting of n distinct (i.e., mutually exclusive) outcomes, s 1, s 2, …, s n. An acceptable probability assignment consists of assigning to each outcome s i a number p i (the probability of s i ) according to these rules: –1. The probability of each outcome is a number between 0 and 1. (PINGTO! and PINN!) 0 <= p 1 <= 1, 0 <= p 2 <= 1, …, 0 <= p n <= 1, –2. The sum of the probabilities of all possible outcomes is 1. p 1 + p 2 + p 3 + … + p n =1 (or p i for short) – Dont forget: 7.4 Further probability notions

Similar presentations

Presentation is loading. Please wait....

OK

AP Statistics From Randomness to Probability Chapter 14.

AP Statistics From Randomness to Probability Chapter 14.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on online shopping cart project in php Ppt on trial and error 1997 Ppt on network theory pdf Ppt on different solid figures pictures Ppt on office ergonomics Ppt on banking sector in pakistan Ppt on aquatic animals Ppt on coordinate geometry for class 9 download Ppt on eye of tiger Mp ppt online form 2012