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 4.3 Toss a coin the sample space is S = {H, T}. Example: From rolling a die, S = {1, 2, 3, 4, 5, 6}. week6 1
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, then A = {HT, TH}. Notation: The probability of an event A is denoted by P(A). week6 2
Union and Intersection of events 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 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. week6 3
Probability rules 1. The probability P(A) of any event A satisfies 0 P(A) 1. 2. If S is the sample space in a probability model, then P(S) = 1. 3. The complement of any event A is the event that A does not occur, written as A c. The complement rule states that P(A c ) = 1 - P(A). 4. 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 week6 4
Venn diagram week6 5
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. week6 6
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. week6 7
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: Color Brown Red Yellow Green Orange Blue Probability 0.30.20.20.10.10? 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? week6 8
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. (a) Find P(A) and P(B). (b) Describe A c in words and find P(A c ) by the complement rule. (c) Describe {A or B} in words and find its probability by the addition rule. week6 9
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 count of outcomes in A P ( A) = = count of outcomes in S Example: count outcomes in k A pair of fair dice are rolled. What is the probability that the 2 nd die lands on a higher value than does the 1 st? of A week6 10
General Addition rule for the unions of two 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). 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? 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? week6 11
Conditional Probability The probability we assign to an event can change if we know that some other event has occurred. When P(A) > 0, the conditional probability that B occurs given the information that A occurs is Example P(B A) = P ( A and B) P( A) Here is a two-way table of all suicides committed in a recent year by sex of the victim and method used. week6 12
(a) 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? (d) 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. week6 13
Independent events 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. week6 14
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? week6 15
General Multiplication Rule The probability that both of two events A and B happen together can be found by P(A and B) = P(A) P(B A) Example 4.33 on page 317 in IPS. 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. week6 16
Bayes s Rule If A and B are any events whose probabilities are not 0 or 1, then PAB ( ) = P( BAPA ) ( ) P( BAPA ) ( ) + PBA ( c) PA ( c) Example: Following exercise using tree diagram. Suppose that A 1, A 2,, A k are disjoint events whose probabilities are not 0 and add to exactly 1. That is any outcome is in exactly one of these events. Then if C is any other even whose probability is not 0 or 1, P( A i C) = P( C A ) P( A ) + P( C A P( C A ) P( A ) i ) P( A ) + + P( C A ) P( A 1 1 2 2 k k i ) week6 17
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. (a) What is the probability that the test shows a positive result? (b) What is the probability that a person selected at random is one who has the disease but was diagnosed healthy? (c) What is the probability that the person is correctly diagnosed and is healthy? (d) If the test shows a positive result, what is the probability this person actually has the disease? week6 18
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? week6 19
Exercise 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. (a)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? week6 20
Question 13 Term Test Summer 99 A space vehicle has 3 o-rings which are located at various field joint locations. Under current wheather conditions, the probability of failure of an individual o-ring is 0.04. (a) 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. week6 21
Question 23 Final exam Dec 98 A large shipment of items is accepted by a quality checker only if a random sample of 8 items contains no defective ones. Suppose that in fact 5% of all items produced by this machine are defective. Find the probability that the next two shipments will both be rejected. week6 22
Question 9 Final exam Dec 2001 You are going to travel Montreal, Ottawa, Halifax, and Calgary, but the order is arbitrary. You put 4 marbles in a box, each one labeled for one city, and draw randomly. The first marble is the first city you will visit, the 2nd marble indicates your 2nd stop etc. What is the probability that you visit Ottawa just before or just after you visit Montreal? week6 23