Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY Chapter 3: Basic Probability Concepts 3.1 General Definitions and Concepts: Probability: Probability is a measure (or number) used to measure the chance of the occurrence of some event. This number is between 0 and 1. An Experiment: An experiment is some procedure (or process) that we do. Sample Space: The sample space of an experiment is the set of all possible outcomes of an experiment. Also, it is called the universal set, and is denoted by Ω. An Event: Any subset of the sample space Ω is called an event. •φ⊆Ω is an event (impossible event) • Ω⊆Ω is an event (sure event) Example: Experiment: Selecting a ball from a box containing 6 balls numbered 1,2,3,4,5 and 6. •This experiment has 6 possible outcomes Ω={1, 2, 3, 4, 5, 6}. •Consider the following events: E1 =getting an event number = 2, 4, 6 }⊆Ω E2 =getting a number less than 4 = 1, 2, 3 }⊆Ω E3 =getting 1 or 3 = 1, 3}⊆Ω E4 =getting an odd number = 1, 3, 5}⊆Ω E5 =getting a negative number = }=φ⊆Ω E6 =getting a number less than 10 = 1, 2,3, 4, 5, 6 }=Ω⊆Ω Notation: n(Ω =)no. of outcomes (elements) in Ω n(E =)no. of outcomes (elements) in the event E INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY Equally Likely Outcomes: The outcomes of an experiment are equally likely if the occurrences of the outcomes have the same chance. Probability of An Event: • If the experiment has N equally likely outcomes, then the probability of the event E is denoted by P(E) and is defined by: P(E =) n E) = n(Ω ) n (E ) no. of outcomes in E = N no. of outcomes in Ω Example: In the ball experiment in the previous example, suppose the ball is selected randomly. Determine the probabilities of the following events: E1 =getting an event number E2 =getting a number less than 4 E3 =getting 1 or 3 Solution: }; n Ω)=6 E1 ={2, 4, 6 }; n(E1 )=3 E2 ={1, 2, 3 }; n(E2 )=3 E3 ={1, 3 }; n(E3 )=2 Ω={1, 2, 3, 4, 5, 6 The outcomes are equally likely . ∴P(E1 =)3 P(E1 =)3 6 , P(E3 =)2 , 6 3.2 Some Operations on Events: Let A and B be two events defined on the sample space Ω. Union of Two events: A ∪B A ∪B Consists of all outcomes in A or in B or in both A and B. A ∪B Occurs if A occurs, or B occurs, or both A and B occur. 6 INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY Intersection of Two Events: A ∩B A ∩B Consists of all outcomes in both A and B. A ∩B Occurs if both A and B occur. Complement of an Event: Ac Ac is the complement of A. Ac consists of all outcomes of Ω but are not in A. Ac occurs if A does not. Example: Experiment: Selecting a ball from a box containing 6 balls numbered 1, 2, 3, 4, 5, and 6 randomly. Define the following events: E1 ={2, 4, 6 }= getting an even number. E2 ={1, 2, 3}= getting a number< 4. INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY E3 ={1, 3 }= getting 1 or 3. E4 ={1, 3, 5}= getting an odd number. (1) E1 ∪E2 ={1, 2, 3, 4, 6 } = getting an even number or a number less than 4 𝑝(𝐸1 ∪ 𝐸2 ) = 𝑛(𝐸1 ∪ 𝐸2 ) 5 = 6 𝑛( Ω) (2) E1 ∪E2 ={1, 2, 3, 4, 6 } = getting an even number or a number less than 4 𝑝(𝐸1 ∪ 𝐸2 ) = 𝑛(𝐸1 ∪ 𝐸4 ) 6 = =1 6 𝑛( Ω) Note: E1 ∪E4 = Ω. E1 and E4 are called exhaustive events. INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY (3) E1 ∩E2 ={2 }= getting an even number and a number less than 4. 𝑛(𝐸1 ∪ 𝐸2 ) 1 𝑝(𝐸1 ∩ 𝐸2 ) = = 𝑛(Ω) 6 E1 ∩E2 =Ø= getting an even number and an odd number. (4) 𝑝(𝐸1 ∩ 𝐸2 ) = 𝑛(𝐸1 ∪ 𝐸4 ) 𝑛(Ø) 0 = = =0 𝑛(Ω) 6 6 Note: E1 ∩E4 =φ. E1 and E4 are called disjoint (or mutually exclusive) events. c E1c = not getting an even number = {2, 4, 6} = {1, 3, 5} (5) = getting an odd number. = E4 Notes: 1. The events A1 , A2 ,K, An are exhaustive events if A1 ∪A2 ∪K∪An =Ω. 2. The events A and B are disjoint (or mutually exclusive) if A ∩B =φ. It is impossible that both events occur together (in the same time). In this case: INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY P(A ∩B =)0 P(A ∪B =)P()A+P B) If A∩B ≠φ, then A and B are not mutually exclusive (not disjoint). A∩B ≠φ A and B are not mutually exclusive (It is possible that both events occur in the same A∩B = φ A and B are mutually exclusive (disjoint) (It is impossible that both events occur in the same time) time) 3. A ∪Ac =Ω, A and Ac are exhaustive events. A ∩Ac =φ, A and Ac are disjoint events. 4. n Ac =n(Ω −)()nA P Ac =1 −P A) 3.3 General Probability Rules: 1. 0 ≤P(A ≤)1 2. 3. 4. 5. P(Ω =)1 P(φ =)0 P Ac =1−P(A ) For any events A and B: 6. P(A ∪B )=P(A +)(PB −)P A ∩B) For disjoint events A and B INTRODUCTION TO BIO STATISTICS ST.PAULS UNIVERSITY 7. P(A ∪B)=P(A +)(PB ) For any events A and B: P(A∩BC)= P(A) −P(A∩B) P(AC∩B)= P(B) −P(A∩B) P(AC∩BC)= 1 −P(A∪B) 8. For mutually exclusive (disjoint) events E1 , E2 ,K, En : P(E1 ∪E2 ∪K∪En )= P E1 )+P E2 )+L+P En ) 3.4 Applications: We will consider several examples. Example: 630 patients are classified as follows : (Simple frequency table) Blood O A B AB (E1 ) E2 ) E3 ) E4 ) Type Total No. of 284 258 63 25 630 patients •Experiment: Selecting a patient at random and observe his/her blood type. •This experiment has 630 equally likely outcomes ∴n(Ω =)630