Sample Spaces • Sample Space – Set of all possible outcomes (or say sample points) of a random experiment. 第二章 Basic Concepts of Probability Theory – Examples: Toss a coin three times and note the sequence of heads and tails • The sample spaces S={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} • Discrete sample space – If S is countable – Ex: Toss a coin three times and note the number of heads • S={0, 1, 2, 3} • Continuous sample space – If S is not countable – Ex: pick a number at random between zero and one • S={x: 0≤ x ≤ 1}=[0,1] 1 3 Events Random Experiments • Subset of interested outcomes contained in the sample space. • An experiment in which the outcome varies in an unpredictable fashion when the experiment is repeated under the same conditions – Ex: the three tosses give the same outcome A={HHH, TTT} • Two special event – Certain event S: consists of all outcomes (always occurs) – Null event ∅ : contains no outcomes and never occurs • Simple event: Unique outcome of an experiment – Ex: Flipping a coin, outcome: Tails • Compound Event: more than one outcome 2 – Ex: the three tosses give the same outcome A={HHH, 4 TTT} Set Theory • The intersection of two events that are common to the two events denoted by the symbol, A ∩ B, or just AB. For example: E=B∩C={1,3} • The union of two events denoted as A ∪ B, or A+B. For example: S C={1,2,3} A A D=B∪C={1,2,3,6} A∪A=S • Two events are said to be mutually exclusive (or say disjoint) if they have no sample points in common – that is, if the occurrence of one event excludes the occurrence of the other. For example: A={1,2} B={1,3,6} A and A are mutually exclusive events (A∩ A = ∅) • In random experiments we are interested in the occurrence of events that are represented by sets (集合) • Definition: universal set U: consists of all possible outcomes, i.e., sample space n 5 • roll of a die. The Sample space is S={1,2,3,4,5,6} ∪ Ak = A1 ∪ A2 ∪ ... ∪ An k =1 7 Venn Diagrams S • In depicting multiple events, Venn diagrams are excellent visual tools. – The six outcomes are the sample points of the experiment. • An event is a subset of S, and may consist of any number of sample points. For example: S A={2,4} A A • The complement of the event A, denoted by A , consists of all the sample points in S that are not in A: S A B A A={1,3,5,6} 6 8 Venn Diagram Example 1 Complement of an Event A • is the subset of all elements of sample space (Ω) that are not in A. • Denoted as A' or Ac • A = Cars with Sunroofs B = Cars with Air conditioning • What does the shaded area represent ? A A B B 9 11 10 12 Venn Diagram Example 2 • A = Cars with Sunroofs B = Cars with Air conditioning • What does the shaded area represent ? A B Corollary 2 Axioms of Probability P[A] ≤1 P(A): the probability of event A Corollary 3 Axiom 1 P ( A) ≥ 0 for any event A P[∅] = 0 Axiom 2 P ( S ) = 1 Corollary 4 If all Ai’s are mutually exclusive, then k Axiom 3 P ( A1 ∪ A2 ∪ ... ∪ Ak ) = ∑ P( Ai ) (finite set) ∞ If A1, A2, …, An are pairwise mutually exclusive n n (兩兩間互斥), then P( ∪ Ak ) = ∑ P[ Ak ] k =1 i =1 for n ≥ 2 k =1 P ( A1 ∪ A2 ∪ ...) = ∑ P( Ai ) (infinite set) i =1 13 15 Corollary 5 Corollary 1: ‘’Additive Rule” of Probability For any event A, P[Ac]=1-P[A] • In the case that two events are not mutually exclusive, we can use the additive rule below: If A and B are mutually exclusive, then P ( A ∩ B ) = 0. P (A ∪ B ) = P (A) + P (B ) − P (A ∩ B ) If any two events A and B, are collectively exhaustive, then P(A+B)= 1 (where “+” means “Union”, i.e. A+B=Sample space) 14 A B 16 Corollary 6 n n P[∪Ak ] = ∑ P[ A j ] − ∑ P[ Aj ∩ Ak ] + k =1 j =1 + (−1) n +1 P[ A1 ∩ j <k Probability Example 1 ∩ An ] • Prob Student has Visa Card = 0.5 • Prob Student has Master Card = 0.15 • Prob Student has Both Cards = 0.1. Corollary 7 If A⊂ B , P[A] <= P[B] A B • What is the probability that a student does not have a Master Card ? • What is the probability that a student has neither card? (Draw a Venn Diagram) 17 Ex. A card is drawn from a well-shuffled deck of 52 playing cards. What is the probability that it is a queen or a heart? 19 Probability Example 2 • Given the following VENN diagram, – What is the prob that events A and B will occur? – What is the prob that events A and B and C will occur? Q = Queen and H = Heart P (Q ) = 4 13 1 , P ( H ) = , P (Q ∩ H ) = 52 52 52 P (Q ∪ H ) = P (Q) + P ( H ) − P(Q ∩ H ) = 4 13 1 + − 52 52 52 = 16 4 = 52 13 18 P P P P P P P P (A) = 0.7 (B) = 0.8 (C) = 0.75 (A or B) = 0.85 (A and B) = ? (A and C) = 0.55 (B and C) = 0.60 (A U B U C) = 0.98 A B C 20 2.2.1 Discrete Sample Space (countable sample space) • For Discrete sample spaces, we can assign probabilities for outcomes (elementary events) • Ex: if sample space S={a1, a2, …, an} event B={a1’, a2’, …, am’} Î P(B)=P[a1’]+P[a2’]+…+P[am’] • If the element in S is equally likely outcomes Î P[a1]=P[a2]=…=P[an]=1/n Î P(B)=P[a1’]+P[a2’]+…+P[am’]=m/n 21 Ex: 2.9 An urn contains 10 identical balls numbered 0, 1,…,9. A random experiment involves selecting a ball from the urn and noting the number of the ball. Find the probability of the following events: A=“number of ball selected is odd” B=“number of ball selected is a multiple of 3,” C=“number of ball selected is less than 5,” and of A∪B and A ∪B ∪C. • Î • Ex:2.10 suppose that a coin is tossed three times. If we observe the sequence of heads and tails, then there are eight possible outcomes S={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. • If we assume that the outcomes of S are equiprobable, then the probability of each of the eight elementary events is 1/8. Î P[“2 heads in 3 tosses”]=P[{HHT, HTH, THH}] =3/8 • If we count the number of heads in three tosses instead of observing the sequence of heads and tails. The sample space is now S={0, 1, 2, 3}. If we assume the outcomes of S to be equiprobable, the each of the elementary events of S has probability ¼ Î P[“2 heads in 3 tosses”]=P[{2}]=1/4 • The two assignments are not consistent with each other Î which one of the assignments is acceptable? 23 • Ex 2.11 A fair coin is tossed repeatedly until the first heads shows up; the outcome of the experiment is the number of tosses required until the first heads occurs. Find the probability law for this experiment Îthe sample space S={1, 2, 3, …}. Suppose the experiment is repeated n times. Let Nj be the number of trials in which the jth toss results in the first heads. If n is very large, we expect N1 to be approximately n/2 since the coin is fair. This implies that a second toss is about n-N1 ≅ n/2 times. Î For large n, the relative frequencies fj ≅ Nj/n = (1/2)j, j = 1, 2, … Î P[j tosses till first heads] = (1/2)j , j = 1, 2, … Î Verify that these probabilities add up to one with α = 1/2 sample space S = {0, 1,…,9} A= {1, 3, 5, 7, 9} B={3, 6, 9} C={0, 1, 2, 3, 4} P[A]=5/10, P[B]=3/10, P[C]= 5/10 P[A ∪B ]=P[A]+P[B]-P[A∩B]=5/10+3/10-P[3,9]=6/10 P[A ∪B ∪C]=P[A]+P[B]+P[C]-P[A ∩B]-P[A ∩C]-P[B ∩C]+P[A ∩B ∩C] =5/10+3/10+5/10-2/10-2/10-1/10+1/10 = 9/10 or X=A ∪ B Î P[A ∪B ∪C]=P[X∪C] =P[X]+P[C]-P[X∩C] ∞ =6/10+5/10-2/10 =9/10 ∑α 22 j =1 j = α 1−α =1 α =1/ 2 24 • Ex: 2.12 Consider the random experiment “pick a number x at random between zero and one.” The sample space S for this experiment is the unit interval [0, 1], which is uncountably infinite. If we suppose that all the outcomes S are equally likely to be selected, then we would guess that the probability that the outcome is in the interval [0, 1/2] is the same as the probability that the outcome is in the interval [1/2, 1]. We would also guess that the probability of the outcome being exactly equal to ½ would be zero since there are an uncountably infinite number of equally likely outcomes. 25 2.2.2 Continuous Sample Space 27 Homeworks • The outcomes are numbers that can assume a continuum of values • The sample space S be the entire real line R • For continuous sample spaces, we can assign probabilities for • 2.1, 2.2, 2.14, 2.15 • 2.21, 2.23, 2.24, 2.33, 2.35 – Intervals of the real line – Rectangular regions in the plane 26 28