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Chapter 6 Probability Probability of Simple Events Probability is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes how likely it is that some event will occur. Probability falls into 3 major approaches. 1.Classical Approach 2.Empirical/Experimental Approach 3.Subjective Approach We will discuss each approach in detail, but first we need to look at some basic ideas associated with probability.

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Probability vs. Statistics: Population known Population unknown Infer sample composition Take sample and infer population In probability, an experiment is any process that can be repeated in which the results are uncertain. Probability experiments do not always produce the same results or outcome, so the result of any single trial of the experiment is not known ahead of time.

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Suppose we are to flip a coin one time, what is the probability that we observe a tails? ½, 0.5, 50/50, 50% So if we flip the coin 10 times, would we definitely see 5 tails? Why not? What if we flipped the coin 100 times? A million times? The Law of Large Numbers As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome.

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Suppose I have a fair die and I am going to roll that die one time and observe the outcome. What are all the possible outcomes? The sample space, S, of a probability experiment is the collection of all possible outcomes or simple events. S = {1,2,3,4,5,6} A simple event is any single outcome from a probability experiment. Each simple event is denoted e i. e 1 = {1}, e 2 = {2}, …, e 6 = {6}, An event is any collection of outcomes from a probability experiment. An event may consist of one or more simple events. Events are denoted using capital letters such as E. E = “roll an odd #” = {1,3,5} F = “roll a # < 4” = {1,2,3}

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Properties of Probabilities We define the probability of an event, denoted P(E), as the likelihood of that event occurring. Probabilities have some properties that must be satisfied. 1. The probability of any event E, P(E), must be between 0 and 1 inclusive. That is, 0 P(E) If an event is impossible, the probability of the event is If an event is a certainty, the probability of the event is If S = {e 1, e 2,..., e n } then P(e 1 ) + P(e 2 ) P(e n ) = 1. We will now discuss the three methods or approaches for determining probabilities. All possible simple events

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Classical Approach The classical method of computing probabilities requires equally likely outcomes. An experiment is said to have equally likely outcomes when each simple event has the same probability of occurring. Some examples would be each number of a die, each card in a deck of cards, and each side of a coin. Computing Probabilities Using the Classical Method If an experiment has n equally likely simple events and if the number of ways that an event E can occur is m, then the probability of E, P(E), is So, if S is the sample space of this experiment, then where N(E) is the number of simple events in E and N(S) is the number of simple events in the sample space.

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Example: Let the sample space be S = {1,2,3,4,5,6,7,8,9,10}. Suppose the simple events are equally likely. Compute the probability of the event E = “an odd number” E = {1,3,5,7,9} N(E) = 5 N(S) = 10

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Empirical/Experimental Approach In this approach, probabilities are obtained from empirical evidence, that is, evidence based upon the outcomes of a probability experiment. Approximating Probabilities through the Empirical Approach The probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the experiment. P(E) relative frequency of E

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Example: On September 8, 1998, Mark McGwire hit his 62 nd homerun of the season. Of the 62 homeruns he hit, 26 went to left field, 21 went to left center, 12 went to center, 3 went to right center and 0 went to right field. 1. What is the probability that a randomly selected homerun was hit to left center field? 2. What is the probability that a randomly selected homerun was hit to right field? 3. Is it impossible for Mark McGwire to hit a homerun to right field? P(“left ctr”) = 21/62 = 0.34 P(“Rt”) = 0/62 = 0 No.

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Tree Diagrams Tree diagrams can be used to determine the sample space of an experiment. Example: Compute the probability of having one boy and three girls in a four-child family assuming boys and girls are equally likely. START 1 st Child 2 nd Child 3 rd Child 4 th Child G B GG G G G B B B B BB B B B G GG G BGBGBGBGBG path: P(B,G,G,G) = (0.5)(0.5)(0.5)(0.5) = = pathways: = 4(0.0625) = 0.25

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Subjective Approach Subjective probabilities are probabilities obtained based upon an educated guess. If you watch the Weather Channel, maybe they say that the chance of rain today is 50%, but the local news says that there is a 75% chance of rain today. These are very different chances for rain. The reason for these differences is because people interpret information differently. Because subjective probabilities are based upon personal judgments, they should be interpreted with extreme skepticism.

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The Addition Rule Complements Compound Events are formed by combining two or more simple events. 1) The probability that both events E and F will occur P(E and F) = P(E F) 2) The probability that either E or F will occur P(E or F) = P(E F) 3) The probability that event E will occur given that event F has already occurred. P(E|F) – read as probability of E given F Intersection Union Conditional Probability

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Let E and F be two events. 1.E and F is the event consisting of simple events that belong to both E and F. 2.E or F is the event consisting of simple events that belong to either E or F or both. Example: Let E = {1,2}; F = {2,3} E ∩ F = {2} E U F = {1,2,3} Let us visualize these concepts using Venn Diagrams. S EF E ∩ F

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Addition Rule For any two events E and F, P(E or F) = P(E F) = P(E) + P(F) – P(E and F) = P(E) + P(F) –P(E F) S EF Counted twice

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S EF If events E and F have no simple events in common or cannot occur simultaneously, they are said to be disjoint or mutually exclusive. Addition Rule for Mutually Exclusive Events If E and F are mutually exclusive events, then P(E or F) = P(E) + P(F) Note: E ∩ F = {Ø} thus,P(E ∩ F) = 0 P(E or F) = P(E) + P(F) –P(E F)

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Complements Suppose the probability of an event E is known and we would like to determine the probability that E does not occur. This can easily be accomplished using the idea of complements. Complement of an Event Let S denote the sample space of a probability experiment and let E denote an event. The complement of E, denoted E C, is all simple events in the sample space S that are not simple events in the event E. Complement Rule If E represents any event and E C represents the complement of E, then P(E C ) = 1 – P(E) S E EcEc

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Example: A standard deck of cards contains 52 cards. One card is randomly selected from the deck. 1. Compute the probability of randomly selecting a two or three from a deck of cards. 2. Compute the probability of randomly selecting a two or three or four from a deck of cards. 3. Compute the probability of randomly selecting a two or club from a deck of cards. 4. Compute the probability of randomly selecting a card other than a two from a deck of cards.

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1. Compute the probability of randomly selecting a two or three from a deck of cards. 2. Compute the probability of randomly selecting a two or three or four from a deck of cards. Note: Mutually exclusive events as above.

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3. Compute the probability of randomly selecting a two or club from a deck of cards. 4. Compute the probability of randomly selecting a card other than a two from a deck of cards.

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The Multiplication Rule Conditional Probability The notation P(F | E) is read “the probability of event F given event E”. It is the probability of an event F occurring given the occurrence of the event E. The Multiplication Rule The probability that two events, E and F both occur is P(E and F)= P(E ∩ F) = P(E) * P(F | E) In words, the probability of E and F is the probability of event E occurring “times” the probability of event F occurring given the occurrence of event E.

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Example: Let S = {1,2,3,4} E = {1,2} F = {2,3} then (E∩F) = {2} What is P(E∩F) ? S EF Note: P(E∩F) is referred to as the joint probability of E and F. P(E) is referred to as the marginal probability of E. Also note: Conditional =

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First draw a tree diagram Example: A bag of 30 tulip bulbs was purchased from a nursery. The bag contains 12 red tulip bulbs, 10 yellow tulip bulbs and 8 purple tulip bulbs. 1. What is the probability that two randomly selected tulip bulbs will both be red? 2. What is the probability that the first bulb selected is red and the second is yellow? 3. What is the probability that the first bulb selected is yellow and the second is red? 4. What is the probability that one bulb is red and the other yellow? Tree on board 12/30 10/30 8/30 11/29 10/29 8/29 12/29 9/29 8/29 STARTRP R Y Y P RP Y RP Y 12/29 10/29 7/29

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1. What is the probability that two randomly selected tulip bulbs will both be red? 2. What is the probability that the first bulb selected is red and the second is yellow? 3. What is the probability that the first bulb selected is yellow and the second is red? 4. What is the probability that one bulb is red and the other yellow?

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Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F. Two events are dependent if the occurrence of event E in a probability experiment affects the probability of event F. From previous example: P(R)=12/30 ≠ P(R|Y) = 12/29 Independent Events Two events E and F are independent if and only if P(F | E) = P(F) or P(E | F) = P(E) Multiplication Rule for Independent Events If E and F are independent events, the probability that E and F both occur is P(E and F) = P(E ∩ F) = P(E)*P(F) In words, the probability of E and F is the probability of event E occurring times the probability of event F occurring.

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Example: 2 fair coinsP(E = “head”) = 0.5 P(F = “tail”) = 0.5 Each toss is an independent event. So P(E ∩ F) = P(E)*P(F) = 0.25 Mutually Exclusive vs.Independent P(E U F) = P(E) + P(F) EF S E S P(E|F) = P(E) = F

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Conditional Probability Conditional Probability Rule If E and F are any two events, then The probability of event F occurring given the occurrence of event E is found by dividing the probability of E and F by the probability of E. Or, the probability of event F occurring given the occurrence of event E is found by dividing the number of simple events in E and F by the number of simple events in E. Likewise,

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Hence, we can use Bayes Rule to conclude, P(E|F)*P(F) = P(F|E)*P(E) = P(E ∩ F) i.e. Multiplication Rule Example: A box contains 100 microchips, some of which were produced by factory 1 and the rest by factory two. Some of the chips are defective and some are good. An experiment consists of choosing one microchip at random from the box and testing whether it is good or defective. The data are presented in the following table. factory 1factory 2 Total defective155 good4535 Total Like a contingency table, but is this probability or statistics? Population is known, therefore probability

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Make a joint probability table: factory 1factory 2 Total defective good Total Joint Probabilities Marginal Probabilities 1. Find the probability of being defective. 2. Find the probability of being made in factory one. 3. Find the probability of being good. 4. Find the probability of being made in factory two. 5. Find the probability of being defective and made in factory one. 6. Find the probability of being defective given made in factory one. 7. Find the probability of made in factory one given defective. 8. Are the events of selecting a defective chip and one made at factory one independent events? 9.Are the events of selecting a defective chip and one made at factory one mutually exclusive events?

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1.Find the probability of being defective. 2. Find the probability of being made in factory one. 3. Find the probability of being good. 4. Find the probability of being made in factory two. factory 1factory 2Total defective good Total

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5. Find the probability of being defective and made in factory one. 6. Find the probability of being defective given made in factory one. 7. Find the probability of made in factory one given defective. factory 1factory 2 Total defective good Total

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factory 1factory 2 Total defective good Total Are the events of selecting a defective chip and one made at factory one independent events? 9. Are the events of selecting a defective chip and one made at factory one mutually exclusive events? No. P(D|Fac1) = 0.25 ≠ 0.20 = P(D) No. P(D∩Fac1) = 0.15 ≠ 0 Note: Joint Tables may be constructed by two means. - Empirical - Theoretical

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