Presentation on theme: "Probability I Introduction to Probability"— Presentation transcript:
1Probability I Introduction to Probability A Satisfactory outcomes vs. total outcomesB Basic PropertiesC TerminologyII Combinatory ProbabilityA The Addition Rule – “Or”The special addition rule (mutually exclusive events)The general addition rule (non-mutually exclusive events)B The Multiplication Rule – “And”The special multiplication rule (for independent events)The general multiplication rule (for non-independent events)
2Probability for Equally Likely Outcomes Suppose an experiment has N possible outcomes, all equally likely. Then the probability that a specified event occurs equals the number of ways, f, that the event can occur, divided by the total number of possible outcomes. In symbolsNumber of ways a given event can occurProbability of a given event =Total of all possible outcomes
3Frequency distribution of annual income for U.S. families
4Probability from Frequency Distributions What is the a priori probability of having an income between $15,000 and $24,999
6Frequency distribution for students’ ages What is the likelihood of randomly selecting a student who is older than 20 but less than 22?What is the likelihood of selecting a student who’s age is an odd number?What is the likelihood of selecting a student who is either 21 or 23?
9Probabilities of 2 throws of the die What is the probability of a 1 and a 3?What is the probability of two sixes?What is the probability of at least one 3?2/361/3612/36
10The Sum of Two Die Tosses What is the probability that the sum will be5?7?What is the probability that the sum will be 10 or more?What is the probability that the sum will be either 3 or less or 11 or more?4/366/366/363/36 + 3/36
11Two computer simulations of tossing a balanced coin 100 times
12Basic Properties of Probabilities Property 1: The probability of an event is always between 0 and 1, inclusive.Property 2: The probability of an event that cannot occur is 0. (An event that cannot occur is called an impossible event.)Property 3: The probability of an event that must occur is 1. (An event that must occur is called a certain event.)
32(a) Two mutually exclusive events (b) Two non-mutually exclusive events
33(a) Three mutually exclusive events (b) Three non-mutually exclusive events (c) Three non-mutually exclusive events
34The Special Addition Rule If event A and event B are mutually exclusive, thenMore generally, if events A, B, C, … are mutually exclusive, thenThat is, for mutually exclusive events, the probability that at least one of the events occurs is equal to the sum of the individual probabilities.
36The General Addition Rule If A and B are any two events, thenP(A or B) = P(A) + P(B) – P(A & B).In words, for any two events, the probability that one or the other occurs equals the sum of the individual probabilities less the probability that both occur.
37P(A or B): Spade or Face Card P (spade) + P (face card) – P (spade & face card)= 1/4 + 3/13 – 3/52 = 22/52
38The Special Multiplication Rule (for independent events) If events A, B, C, are independent, thenP(A & B & C & ¼) = P(A) · P(B) · P(C)¼.What is the probability of all of these events occurring:Flip a coin and get a headDraw a card and get an aceThrow a die and get a 1P(A & B & C ) = P(A) · P(B) · P(C) = 1/2 X 1/13 X 1/6
39Conditional Probability: For non-independent events The probability that event B occurs given that event A has occurred is called a conditional probability. It is denoted by the symbol P(B | A), which is read “the probability of B given A.” We call A the given event.
41Contingency table for age and rank of faculty members (using frequencies)
42The Conditional-Probability Rule If A and B are any two events, thenIn words, for any two events, the conditional probability that one event occurs given that the other event has occurred equals the joint probability of the two events divided by the probability of the given event.
44Joint probability distribution (using proportions) P( R3 | A4 ) == 0.031/0.217= 0.142P( A4 | R3 ) == 0.031/.0275= 0.112
45Contingency table of marital status and sex (using proportions)
46Joint probability distribution (using proportions)
47The General Multiplication Rule If A and B are any two events, thenP(A & B) = P(A) · P(B | A).In words, for any two events, their joint probability equals the probability that one of the events occurs times the conditional probability of the other event given that event.Note: Either1) The events are independent and then P(A & B) = P(A) · P(B).Or2) The events are not independent and then a contingency table must be used
48Independent EventsEvent B is said to be independent of event A if the occurrence of event A does not affect the probability that event B occurs. In symbols,P(B | A) = P(B).This means that knowing whether event A has occurred provides no probabilistic information about the occurrence of event B.Class Fr So Ju SeMale | 180Female | 360| 540
49Probability and the Normal Distribution What is the probability of randomly selecting an individual with an I.Q. between 95 and 115? Mean 100, S.D. 15.Find the z-score for 95 and 115 and compute the area between
50More Preview of Experimental Design Using probability to evaluate a treatment effect. Values that are extremely unlikely to be obtained from the original population are viewed as evidence of a treatment effect.
51A Preview of Sampling Distributions What is the probability of randomly selecting a sample of three individuals, all of whom have an I.Q. of 135 or more?Find the z-score of 135, compute the tail region and raise it to the 3rd power.This concept is critical to understanding future conceptsSo while the odds chance selection of a single person this far above the mean is not all that unlikely, the odds of a sample this far above the mean are astronomicalz = 2.19P ==XX
52SummaryFor multiple events there are two rules:“AND” (multiplication) and “OR” (addition)There are just a few special considerations:For the “And” rule, if the events are not independent, you don’t multiply, you use a table.For the “Or” rule, if the events are not mutually exclusive you have to subtract off their double count