2.4. Conditional Probability

Size: px

Start display at page:

Download "2.4. Conditional Probability"

Bruce Houston
6 years ago
Views:

1 2.4. Conditional Probability Objectives. Definition of conditional probability and multiplication rule Total probability Bayes Theorem Example (#46 p.80 textbook) Suppose an individual is randomly selected from the population of all adult males living in the United States. Let A be the event that the selected individual is over 6 ft in height, and let B be the event that the selected individual is a professional basketball player. Which do you think is larger, the individual being more than 6 feet tall, knowing P that the individual is a professional basketball player or P the probability of the individual being a professional basketball player, knowing that the individual is more than 6 feet tall Answer with reasoning. Let A be that the individual is more than 6 feet tall. Let B be that the individual is a professional basketball player. P(A B) = the probability of the individual being more than 6 feet tall, knowing that the individual is a professional basketball player. P(B A) = the probability of the individual being a professional basketball player, knowing that the individual is more than 6 feet tall. Because most professional basketball players are tall, so the probability of an individual in that reduced sample space being more than 6 feet tall is very large. On the other hand, the number of individuals that are pro basketball players is small in relation to the number of males more than 6 feet tall P(A B) P(B A) Example What the probability that when rolling a die the side 6 will be on the top if it is known that occurred an even number? Answer

Definition of Conditional Probability 2.4.3. Conditional probability of an even A is the probability of A computed with information that another event B occurred. Notation: P A B Formal definition.

2 Definition of Conditional Probability Conditional probability of an even A is the probability of A computed with information that another event B occurred. Notation: P A B Formal definition. For any two events A and B, PB 0, the conditional probability of A given that B has occurred is defined by P A B P A B P B Example The population of a particular country consists of three ethnic groups. Each individual belongs to one of the four major blood groups. The accompanying joint probability table gives the proportions of individuals in the various ethnic group blood group combinations. PC R and PC Suppose that an individual is randomly selected from the population, and define A type A selected, B type B selected, and events C ethnic group 3 selected a) Calculate P A, PC and P A C b) Calculate both P A C and PC A and explain in context what each PC R and PC probability represents. 2

3 Remark about Contingency Table. In the problem given data is organized in so-called contingency table. The contingency table is the method of organizing data for qualitative variables as a table in which every cell represents one of the categories of a twodirectional classification. In the example above sample consists of elements that have two features (specified blood type and specified ethnic group). The table contains proportions of elements that have different combinations of two variables, blood type and ethnic group. For example, the number (first column, first row) is the proportion in the sample of people with blood type O and classified as the ethnic group 1. Solution. P A ; PC P A C P A C type A selected given ethnic group ; P C A ethnic group 3 given type A selected Multiplication Rule. Example Two cards are drawing one after one from a deck of 52 cards. What is the probability that both cards will be aces? Solution Pfirst ace and second ace I is an ace II is an ace 4 aces 48 others 3

2.4.8. The Law of Total Probability. Consider a system A,..., 1 A k - mutually exclusive and exhaustive events.

.. An, Ai Aj for any i and j, i j, i, j 1, k The probability of any other event B (related to the same sample space) can by found

If in a sample space is present a system of mutually exclusive and exhaustive events (hypotheses), then any event B connected

4 The Law of Total Probability. Consider a system A,..., 1 A k - mutually exclusive and exhaustive events. This means that mutually exclusive events A,..., 1 A k make partition of the sample space : A1 A2... An, Ai Aj for any i and j, i j, i, j 1, k The probability of any other event B (related to the same sample space) can by found by so-called the Formula for Total Probability 1 1 k k P B P A P B A P A P B A k i1 i i P A P B A Idea of Proof. If in a sample space is present a system of mutually exclusive and exhaustive events (hypotheses), then any event B connected with this experiment will take place only together with one of the hypotheses. So, find the probability of the event B means find the probability of the union of mutually exclusive events 4

5 Probability Tree Diagram. When applying the Formula for Total Probability, it is very helpful to construct socalled Probability Tree Diagram. In a Probability Tree Diagram: first set of branches (arrows) presents mutually exclusive events A,..., 1 A k (often called hypotheses ). from the end of each branch (arrow) in first set we draw second level of branches (arrows) in correspondence with options {B given that A occurred} along each arrow is written corresponding probability. number of branches at each level can be arbitrary, but the sum of all probabilities at each level that go from the same node is always equal to 1. i Example Consider taking two balls, one after one, from the box that contains 2 blue balls and 3 red balls. Construct the probability tree for this experiment. First draw Probabilities of hypotheses P Ai, i 1,2 Second draw Conditional probabilities i and PB A i P B A with each hypothesis Example of computing. Pretend the event of interest B is {the ball in the second draw is blue} P B P A P B A P A P B A P B Hint. We were doing multiplication along branches that have blue ball at the end. 5

6 Example % of the students enrolled in a business statistics course had previously taken a finite math course. 30 % of these students received 4.0 grade for the statistics course, whereas 20% of other students received 4.0 grade for the statistics course. a) Draw a tree diagram and label it with all appropriate probabilities. b) What is the probability that a student selected at random receive 4.0 in the statistics course? To find the probability of the event B {student recieved 4 points}, we need choose sequences of arrows that end up with 4.0, multiply probability along each composite branch and add all such products. P{student recieved 4.0}

2.4.16. Bayes Theorem. The Formulation of Bayes Theorem.

Given, that the event B occurred, we don t need any more to consider the entire sample space.

To find j for P A B for any A, we need to count only outcomes that are common j Aj and B and divide this number by number

Only 1 in 100 adults is afflicted with a rare disease for which a diagnostic test has been developed.

7 Bayes Theorem. The Formulation of Bayes Theorem. Let be mutually exclusive and exhaustive events with prior probabilities Then for any other event, Idea of Proof. Given, that the event B occurred, we don t need any more to consider the entire sample space. You will deal only with its part B. To find j for P A B for any A, we need to count only outcomes that are common j Aj and B and divide this number by number of outcomes in B. Example (Example 2.31 p.79 textbook, modified). Incidence of a rare disease. Only 1 in 100 adults is afflicted with a rare disease for which a diagnostic test has been developed. The test is such that when an individual actually has the disease, a positive result will occur 99% of the time, whereas an individual without the disease will show a positive test result only 0.4% of the time. If a randomly selected individual is tested and the result is positive, what is the probability that the individual has the disease? 7

8 Solution. For solving the problem, we will use the Bayes formula. 1) Make a Probability Tree Diagram. 2) In the denominator of Bayes formula must be written P test is positive, computed by Total Probability Formula. P test is positive ) In the numerator of Bayes formula, we have to put the product that corresponds with branch Disease Positive ) Finally, Phas disease given, test positive

9 2.5. Independence. Objectives. Definition of independence of two events Multiplication rule of probability of intersection of two events Independence of more than two events Definition of Independence of Two Random Events. Two events A and B are called independent if and only if P A B P A PB It can be easy shown that for independent events A and B Example2.5.2., P A B P A P A B P A Independence of two events in the probability theory means that the chances of occurrence of one even do not change when the info about occurrence of another event is provided. You toss a coin and it comes up H two times. What is the probability that the next toss will also be a "H"? Answer. 1 2 Example Are two mutually exclusive events independent or no? Answer. No Example Consider throwing two fair dice, red and green. Are the following events independent or dependent? Give reasons for your answers. 1) A = {the sum is 4} and B = {Both dice show the same number} 2) A = {the sum is 6} and B = {The red die shows an even} 9

10 Solution. A the sum is 4 1) {sum is 4}={(1,3), (2,2), (3,1)} P Pthe sum is 4 dice show the same Conditional probability Unconditional probability 1 12 A and B are dependent events 2) A {sum is 6}={(5,1), (4, 2), (3, 3), (2, 4), (1, 5)} P A 5 36 P 3 A B Conditional probability Unconditional probability A and B are dependent events 10

11 Example An ice cream company wishes to use red dye to enhance the color in its strawberry ice cream. The Food and Drug Administration (FDA) requires the dye to be tested for cancer-producing potential in a laboratory using laboratory rats. The results of one test on 1,000 rats are summarized in the following table: a) Convert the table into the proportions table by dividing each entry by 1,000. b) What is the probability that a rat develops cancer, given that this rat ate red 60 3 dye? PC R c) Are developing cancer and eating dye independent events? Developed Cancer C PC, PC R PC No Cancer C events are not independent d) Should the FDA approve or ban the use of the dye? P C R P C. Explain why or why not using Totals Ate Red Dye R Did Not Eat Red Dye R Totals ,000 Developed Cancer C No Cancer C Totals Ate Red Dye R Did Not Eat Red Dye R Totals FDA should ban the use of Red Dye because eating this substance increases Probability of developing cancer. 11

12 An Important Application of Bayes Formula (for reading). Computing the probability that a message containing a given word is spam. Let's suppose the suspected message contains the word "replica". Most people who are used to receiving know that this message is likely to be spam, a proposal to buy counterfeit copies of well-known brands of watches. The spam detection software, however, does not "know" such facts; all it can do is to compute probabilities. The formula used by the software to determine that, is derived from Bayes' theorem P P S P W S S W P S P W S P H P W H where: PS W is the probability that a message is a spam, knowing that the word "replica" is in it; P S is the overall probability that any given message is spam; PW S is the probability that the word "replica" appears in spam messages; PH is the overall probability that any given message is not spam (is "ham"); PW H is the probability that the word "replica" appears in ham messages. The spamicity of a word. Recent statistic shows that the current probability of any message being spam is 80%. Thus, PH P S 0.8; 0.2 However, most Bayesian spam detection software makes the assumption that there is no a priori reason for any incoming message to be spam rather than ham, and considers both cases to have equal probabilities of 50%: PH P S 0.5;

13 The filters that use this hypothesis are said to be "not biased", meaning that they have no prejudice regarding the incoming . This assumption permits simplifying the general formula to: PW S P S W P W S P W H This is functionally equivalent to asking, "what percentage of occurrences of the word "replica" appear in spam messages?" The quantity PW S P S W P W S P W H is called "spamicity" (or "spaminess") of the word "replica", and can be computed. P W S used in this formula is approximated to the frequency of The number messages containing "replica" in the messages identified as spam during the P W H is approximated to the frequency of messages learning phase. Similarly, containing "replica" in the messages identified as ham during the learning phase. For these approximations to make sense, the set of learned messages needs to be big and representative enough. It is also advisable that the learned set of messages conforms to the 50% hypothesis about repartition between spam and ham, i.e. that the datasets of spam and ham are of same size. Of course, determining whether a message is spam or ham based only on the presence of the word "replica" is error-prone, which is why Bayesian spam software tries to consider several words and combine their spamicities to determine a message's overall probability of being spam. 13

Probability deals with modeling of random phenomena (phenomena or experiments whose outcomes may vary)

Probability deals with modeling of random phenomena (phenomena or experiments whose outcomes may vary) Chapter 14 From Randomness to Probability How to measure a likelihood of an event? How likely is it to answer correctly one out of two true-false questions on a quiz? Is it more, less, or equally likely