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BIOL499A (11)
Chapter 14-16

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Department
Biology (Biological Sciences)
Course
BIOL499A
Professor
Blaine Mullins
Semester
Winter

Description
Chapters 14 - 16: Probability In these chapters, we will talk about  Sample space,  Events,  Probability for some events,  Complement of an event,  The union of two events,  The intersection of two events,  Mutually exclusive or disjoint events,  Probability rules,  Conditional probabilities,  Independence of two events,  Random variables,  Probability distribution of a discrete random variable,  Expected value or mean of a discrete random variable. Definition: A probability experiment is any action for which an outcome cannot be predicted with certainty. HO2 2 0 Toss a coin: H = Head or T = Tail Flip a die: 1, 2, 3, 4, 5, 6 Choose a student and record his/her weight: Definition: The set of all possible outcomes is called thesample space. Example 1: Find the sample space for the following probability experiments: (a) Toss a coin:,T} (b) Flip a die:2,3,4,5,6} (c) Toss a coin twice: S={(H,H), (H,T), (T,H), (T,T)} ={HH,HT,TH,TT} (d) Flip a die twice: (1,1), (1, 2),(1,3), (1, 4),1,5), (1,6)   (2,1),(2,2),(2,3),(2,4),(2,5),(2,6)   S             (6,),(6,2),(6,3),(6,4),(6,5,(6,6) (e) Toss a coin until we get a head: Definition: An event is a subset of outcomes from the sample space. An individual outcome from the sample space is called a simple event or an elementary event. e.g.: Flip a die:= {1,2,3,4,5,6} A = {2} B = Odd numbers = {1,3,5} C = Even numbers = {2,4,6} D = Prime numbers = {2,3,5} Definition: The probability of an event is a numerical value that represents the proportion of times that the even t is expected to occur when the experiment is repeated under identical conditions. e.g.: Toss a coin S ={H, T} P({H}) = 1/2 and P(T) = 1/2 For an equally likely model: Number of elements in A P(A ) Number of elements in Sor any set A Example 2: Toss a coin twice and record the outcome heaH) or tailT( ) for each toss. Find the probability of the following events: S = {HH, HT, TH,TT} A = Exactly one head = {HT, TH} B = No head = {TT} C = At least one head = {HT,TH, HH} D = At most one head = {TT, HT, TH} E = Both heads = {HH} Example 3: Flip a die twice. Find the probability that the sum of two numbers is seven. Solution: A = sum is 7 = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} Example 4: A restaurant serves three types of pasta (Spaghetti, Rigatoni, and Fettuccine) with one of two sauces (Tomato or White). A customer will order a pasta dish. How should the owner go about determining the probability that a customer will order A = A spaghetti, B = A white-sauce dish. Example 5: An insurance company checked its records for a recent year and found that of 12299 automobile insurance policies in effect, 2073 made a claim. Among insured derivers under age 25, there were 1053 claims out of 5192 policies. Find the probability that (a) an insured driver will make a claim. (b) an insured driver under 25 years will make a claim. Note: For any probability model, the following two conditions must hold: a) For any event A, we have 0 ▯ P(A) ▯ 1 PE() 1 b)  E is simple Therefore, for examples: a) It is possible to have a coin with P({H}) = 0.7 and P(T) = 0.3. But: b) It is impossible to have a coin with P({H}) = 0.7 and P(T) = 0.4. OR c) It is impossible to have a coin with P({H}) = 1.2 and P(T) = ▯0.2. Example 6: Is it possible to have a die with 1 2 P(1) P) (5)Pad (2) (4) (6) ? 9 9 Solution: a) All probabilities are between zero and 1 b) P(1)P)P (3) (4) (5) (6) 1 2+1 2 1+2 + + + =1 9 9 9 9 9 9 Then A = Odd numbers = {1,3,5} B = Even numbers = {2,4,6} C = Prime numbers = {2,3,5} Definition: The intersection of tAandvB, denotedA B, is the set of all elementary outcomesAthatBare inoccurrence of A Bmeans that bAtandBoccur. e.g.: Flip a die:,3,4,5,6}  A  n numbers 2,4,6    B {2}   Prime numbers  ,3,5   A  Even numbers 2,4,6   AC  {} C  Odd numbers  1,5   Definition: Two evAnandB are mutually exclusive or disjoint when they cannot occur simultaneously. Definition: Two eventsA and B are independent when the occurrence (or the nonoccurrence) ofA does not affect the probability of the occurrence of B and vice versa. It can be shown that two events A and B are independent, if and only if, we hP(A B)  P(A)P(B). Example 7: Toss a coin twice. Let A = First coin is head. B = Second coin is head, and C = At least one head (a) Are A and B independent? How about disjoint? (b) Are A and C independent? How about disjoint? Solution: S = {HH,HT,TH,TT} (a) A HH HT,    A BHH { }    HTH, P( )   PAPB )() (b)  A HH ,HT     A TCHH { , } C    H,TH , PA( )( C PA )(P)C Definition: The union of two events A and B , denoted byA B , is the set of all elementary outcomes that are iA B , or both. The occurrence of A B means that either A or B or both occur. e.g.: Flip a die: S={1,2,3,4,5,6} A  Even numbers 2,4,6    AB {2,4,6}{2,3,5} {2,4,6,2,3,5} {2,3,4,5,6} B Prime numbers ,3,5 A  Even numbers 2,4,6 C  Odd numbers ,3,5  AC ,6  1,3,5 {2,4,6,1,3,5} {1,2,3,4,5,6}   Probability Rule: For any two AvandB, we have: P(A B)  P(A)  P(B)  P(A B) Note thaP(A B)  P(A)  P(B) and only if A and B are disjoint. Example 8: Flip a die twice. Let A = Sum of two numbers is seven, and B = First die is 2. Find probaA B.y of Solution: A = sum is 7 = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} B = First is 2 = {(2,1),(2,2), 2,3), (2,4),(2,5), (2,6)}   , , ,  ,1       AB  2,1 , 2,2 ,(2,3 , 2,4 , , 25, 2,6         , 6,     , , 2,, (2,3),,4 , , 2,6  Example 9: An urn contains 10 balls labeled 1,2,,10 . Suppose one ball is drawn at random from the urn and consider the following events: A = The number is an even number, B = The number is among the first five numbers, C = The number is among the last three numbers. (a) Find the following probabilities: P(A B) , P(AC) P(B C) P(B C) (b) Are A and B independent? How about disjoint? (c) Are B and C independent? How about disjoint? C Definition: The complement of an event A, denoted by A Aor
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