Probability and Sample Space We call a phenomenon random if individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions. The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, probability is a long-term relative frequency. Example: Tossing a coin: P(H) = ? The sample space of a random phenomenon is the set of all possible outcomes. Example: Toss a coin the sample space is S = {H, T}. Example: From rolling a die, S = {1, 2, 3, 4, 5, 6}. STA286 week2

Events An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space. Example: Take the sample space (S) for two tosses of a coin to be the 4 outcomes {HH, HT, TH TT}. Then exactly one head is an event, call it A, and A = {HT, TH}. Notation: The probability of an event A is denoted by P(A). STA286 week2

Union and Intersection of Events The intersection of any collection of events is the event that all of the events occur. Example: The event {A and B} is the intersection of A and B, it is the event that both A and B occur. The union of any collection of events is the event that at least one of the events in the collection occurs. Example: The event {A or B} is the union of A and B, it is the event that at least one of A or B occurs (either A occurs or B occurs or both occur). The compliment of an event A is the set of all elements in S that are not in A. STA286 week2

Probability Rules The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1. If S is the sample space in a probability model, then P(S) = 1. 3. The complement rule states that P(A’) = 1 - P(A) . Two events A and B are disjoint if they have no outcomes in common and so can never occur together. If A and B are disjoint then P(A or B) = P(A U B) = P(A) + P(B) . This is the addition rule for disjoint events and can be extended for more than two events STA286 week2

Venn Diagram Events, relationship between events and their probabilities are best described with vann diagrams. Example…. STA286 week2

Question Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement about an event. (The probability is usually a much more exact measure of likelihood than is the verbal statement.) 0 ; 0.01 ; 0.3 ; 0.6 ; 0.99 ; 1 (a) This event is impossible. It can never occur. (b) This event is certain. It will occur on every trial of the random phenomenon. (c) This event is very unlikely, but it will occur once in a while in a long sequence of trials. (d) This event will occur more often than not. STA286 week2

Probabilities for Finite Number of Outcomes The individual outcomes of a random phenomenon are always disjoint. So the addition rule provides a way to assign probabilities to events with more then one outcome. Assign a probability to each individual outcome. These probabilities must be a number between 0 and 1 and must have sum 1. The probability of any event is the sum of the probabilities of the outcomes making up the event. STA286 week2

Question If you draw an M&M candy at random from a bag of the candies, the candy you draw will have one of six colors. The probability of drawing each color depends on the proportion of each color among all candies made. (a) The table below gives the probability of each color for a randomly chosen plain M&M: What must be the probability of drawing a blue candy? (b) What is the probability that a plain M&M is any of red, yellow, or orange? (c) What is the probability that a plain M&M is not red? Color  Brown Red Yellow Green Orange Blue Probability 0.30 .20 .10 ? STA286 week2

Question Choose an American farm at random and measure its size in acres. Here are the probabilities that the farm chosen falls in several acreage categories: Let A be the event that the farm is less than 50 acres in size, and let B be the event that it is 500 acres or more. Find P(A) and P(B). (b) Describe Ac in words and find P(Ac) by the complement rule. Describe {A or B} in words and find its probability by the addition rule. STA286 week2

Equally Likely Outcomes If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1/k. The probability of any event A is Example: A pair of fair dice are rolled. What is the probability that the 2nd die lands on a higher value than does the 1st ? STA286 week2

General Addition Rule for the Unions of Events If events A and B are not disjoint, they can occur together. For any two events A and B P(A or B) = P(A U B) = P(A) + P(B) - P(A and B). This rule can be extended to more then two events. STA286 week2

Exercise A retail establishment accepts either the American Express or the VISA credit card. A total of 24% of its customers carry an American Express card, 61% carry a VISA card, and 11% carry both. What percentage of its customers carry a card that the establishment will accept? Solution…. STA286 week2

Exercise Among 33 students in a class, 17 earned A’s on the midterm exam, 14 earned A’s on the final exam, and 11 did not earn A’s on either examination. What is the probability that a randomly selected student from this class earned A’s on both exams? Solution…. STA286 week2

Conditional Probability The probability we assign to an event can change if we know that some other event has occurred. Example… When P(A) > 0, the conditional probability that B occurs given the information that A occurs is STA286 week2

Example Here is a two way table of all suicides committed in a recent year by sex of the victim and method used. STA286 week2

What is the probability that a randomly selected suicide victim is male? (b) What is the probability that the suicide victim used a firearm? (c) What is the conditional probability that a suicide used a firearm, given that it was a man? Given that it was a woman? Describe in simple language (don't use the word “probability”) what your results in (c) tell you about the difference between men and women with respect to suicide. STA286 week2

Independent Events P(B | A) = P(B) . P(A and B) = P(A)·P(B) . Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. That is, if A and B are independent then, P(B | A) = P(B) . Multiplication rule for independent events If A and B are independent events then, P(A and B) = P(A)·P(B) . The multiplication rule applies only to independent events; we can not use it if events are not independent. This rule can be extended to more then two events. STA286 week2

Example The gene for albinism in humans is recessive. That is, carriers of this gene have probability 1/2 of passing it to a child, and the child is albino only if both parents pass the albinism gene. Parents pass their genes independently of each other. If both parents carry the albinism gene, what is the probability that their first child is albino? If they have two children (who inherit independently of each other), what is the probability that (a) both are albino? (b) neither is albino? (c) exactly one of the two children is albino? If they have three children (who inherit independently of each other), what is the probability that at least one of them is albino? STA286 week2

General Multiplication Rule The probability that both events A and B happen together can be found by P(A and B) = P(A)· P(B | A) Example 29% of Internet users download music files and 67% of the downloaders say they don’t care if the music is copyrighted. The percent of Internet users who download music (event A) and don’t care about copyright (event B) is P(A and B) = P(A)· P(B | A) = 0.29·0.67 = 0.1943. STA286 week2

Theorem of Total Probability Definition: A partition of the sample space S is a countable collection of events such that Theorem: If is a partition of S such that then STA286 week2

Bayes’s Rule If A and B are any events whose probabilities are not 0 or 1, then Example: Following exercise using tree diagram. Suppose that A1, A2,…, Ak are disjoint events whose probabilities are not 0 and add to exactly 1. That is, A1, A2,…, Ak is a partition of the sample space. Then if C is any other even whose probability is not 0 or 1, then STA286 week2

Exercise The fraction of people in a population who have a certain disease is 0.01. A diagnostic test is available to test for the disease. But for a healthy person the chance of being falsely diagnosed as having the disease is 0.05, while for someone with the disease the chance of being falsely diagnosed as healthy is 0.2. Suppose the test is performed on a person selected at random from the population. What is the probability that the test shows a positive result? What is the probability that a person selected at random is one who has the disease but was diagnosed healthy? What is the probability that the person is correctly diagnosed and is healthy? If the test shows a positive result, what is the probability this person actually has the disease? STA286 week2

Exercise An automobile insurance company classifies drivers as class A (good risks), class B (medium risks), and class C (poor risks). Class A risks constitute 30% of the drivers who apply for insurance, and the probability that such a driver will have one or more accidents in any 12-month period is 0.01. The corresponding figures for class B are 50% and 0.03, while those for class C are 20% and 0.10. The company sells Mr. Jones an insurance policy, and within 12 months he had an accident. What is the probability that he is a class A risk? STA286 week2

Exercise approximately as follows: 37% type A, 13% type B, 44% type The distribution of blood types among white Americans is approximately as follows: 37% type A, 13% type B, 44% type O, and 6% type AB. Suppose that the blood types of married couples are independent and that both the husband and wife follow this distribution. An individual with type B blood can safely receive transfusions only from persons with type B or type O blood. What is the probability that the husband of a woman with type B blood is an acceptable blood donor for her? (b) What is the probability that in a randomly chosen couple the wife has type B blood and the husband has type A? (c) What is the probability that one of a randomly chosen couple has type A blood and the other has type B? (d) What is the probability that at least one of a randomly chosen couple has type O blood? STA286 week2

Question from Past Term Test A space vehicle has 3 ‘o-rings’ which are located at various field joint locations. Under current whether conditions, the probability of failure of an individual o-ring is 0.04. A disaster occurs if any of the o-rings should fail. Find the probability of a disaster. State any assumptions you are making. (b) Find the probability that exactly one o-ring will fail. STA286 week2

Random Variables Example: We roll a fair coin 4 times. Suppose we are interested in the number of heads in the 4 rolls. Let X = number of 5’s. Then X could be 0, 1, 2, 3, 4. X = 0 corresponds to the 1 elements of our 16 elements of the sample space S. X = 1 corresponds to the 4 elements etc. X is an example of a random variable. A random variable is a function that maps a real number to each element in the sample space. Probability models are often stated in terms of random variables. E.g. - model for the # of H’s in 10 flips of a coin. - model for the height of a randomly chosen person. - model for size of a queue. STA286 week2

Discrete Random Variable Definition: A random variable X is said to be discrete if it can take only a finite or countable infinite number of distinct values. A discrete random variable X maps the sample space S onto a countable set. Define a probability mass function (pmf) or frequency function on X such that Where the sum is taken over all possible values of X. The probability distribution of a discrete random variable X is represented by a formula, a table or a graph which provides the list of all possible values that X can take and the pmf for each value. STA286 week2

Examples of Discrete Random Variables 1. Consider the experiment of tossing a coin. Define a random variable as follows: X = 1 if a H comes up = 0 if a T comes up. This is an example of a Bernoulli r.v. The Probability function of X is given in the following table X P(X = x) 1 p 1- p STA286 week2

The probability function of X is given in the following table. Let X be a r.v counting the number of girls in a family with 3 children. The probability function of X is given in the following table. Toss a coin 4 times. Let X be the number of H’s. Find the probability function of X. Draw a probability histogram. Toss a coin until the 1st H. Let X be the number of T’s before the 1st H. Find the probability function of X. x P(X = x) 1 2 3 (0.5)3 = 0.125 3·(0.5)3 = 0.375 STA286 week2

Distribution Function of Random Variables A cumulative distribution function (cdf) of a random variable X is a function defined by If X is a discrete random variable with pmf for x = 0, 1, 2, … then where is the greatest integer ≤ v. Example… STA286 week2

Properties of Distribution Function F is monotone, non decreasing i.e. F(x) ≤ F(y) if x ≤ y. As x  - ∞ , F(x)  0 As x  ∞ , F(x)  1 F(x) is continuous from the right For a < b Why? STA286 week2

Continuous Random Variable - Introduction If the sample space contains an infinite number of possible outcomes equal to the number of points of a line segment, it is called a continuous sample space. A continuous random variable X takes all values in an interval of numbers. The probability distribution of X is often described by a density curve. It is the plot of the probability density function. The total area under a density curve is 1. The probability of any event is the area under the density curve and above the value of X that make up the event. STA286 week2

Example The density function of a continuous r. v. X is given in the graph below. Find 1) P(X < 7) 2) P(6 < X < 8) 3) P(X = 7) 4) P(5.5 < X < 7 or 8 < X < 9) STA286 week2

Formal Definition of Continuous Random Variable A random variable X is continuous if its cumulative distribution function can be written in the form for some non-negative function f. fX(x) is the (Probability) Density Function of X. The Density Function fX(x) of X is simply the derivative of the cdf (if it exists). STA286 week2

Facts and Properties of Pdf Properties of Probability Density Function (pdf) Any function satisfying these two properties is a probability density function (pdf) for some random variable X. Note: fX (x) does not give a probability. For continuous random variable X with density f STA286 week2

Example Kerosene tank holds 200 gallons; The model for X the weekly demand is given by the following density function Check if this is a valid pdf. Find the cdf of X. STA286 week2