Chapter 1 Discrete Probability Distributions - DIM

1.2. DISCRETE PROBABILITY DISTRIBUTIONS 25First toss Second toss Third toss OutcomeHω1HTω2HTHTωω34(Start)THTHTHωωω567Tω8Figure 1.8: Tree diagram for three tosses of a coin.Let A be the event “the first outcome is a head,” and B the event “the secondoutcome is a tail.” By looking at the paths in Figure 1.8, we see thatP (A) =P(B)= 1 2 .Moreover, A ∩ B = {ω 3 ,ω 4 }, and so P (A ∩ B) =1/4.Using Theorem 1.4, we obtainSince A ∪ B is the 6-element set,P (A ∪ B) = P(A)+P(B)−P(A∩B)= 1 2 +1 2 −1 4 = 3 4 .A ∪ B = {HHH,HHT,HTH,HTT,TTH,TTT} ,we see that we obtain the same result by direct enumeration.✷In our coin tossing examples and in the die rolling example, we have assignedan equal probability to each possible outcome of the experiment. Corresponding tothis method of assigning probabilities, we have the following definitions.Uniform DistributionDefinition 1.3 The uniform distribution on a sample space Ω containing n elementsis the function m defined bym(ω) = 1 n ,for every ω ∈ Ω.✷