Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

CHAP. 7] PROBABILITY 125 EXAMPLE 7.2 (Pair of dice) Toss a pair of dice and record the two numbers on the top. There are six pos ...

126 PROBABILITY [CHAP. 7 7.3Finite Probability Spaces The following definition applies. Definition 7.1: LetSbe a finite sample s ...

CHAP. 7] PROBABILITY 127 Theorems on Finite Probability Spaces The following theorem follows directly from the fact that the pro ...

128 PROBABILITY [CHAP. 7 Fig. 7-3 SupposeSis an equiprobable space, andn(A) denotes the number of elements inA. Then: P(A∩E)= n( ...

CHAP. 7] PROBABILITY 129 Multiplying both sides byP (A)gives us the following useful result: Theorem 7.6 (Multiplication Theorem ...

130 PROBABILITY [CHAP. 7 Also, P(A∩B)=P({HHH,HHT})=^14 ,P(A∩C)=P({HHT})=^18 ,P(B∩C)=P({HHT,THH})=^14 Accordingly, P (A)P (B) =^1 ...

CHAP. 7] PROBABILITY 131 Repeated Trials with Two Outcomes, Bernoulli Trials, Binomial Experiment Now consider an experiment wit ...

132 PROBABILITY [CHAP. 7 7.7Random Variables LetSbe a sample of an experiment. As noted previously, the outcome of the experimen ...

CHAP. 7] PROBABILITY 133 The set of ordered pairs(x 1 ,f(x 1 )),(x 2 ,f(x 2 )),...,(xt,f(xt))is called thedistributionof the ran ...

134 PROBABILITY [CHAP. 7 EXAMPLE 7.15 (a) Suppose a fair coin is tossed six times. The number of heads which can occur with thei ...

CHAP. 7] PROBABILITY 135 Binomial Distribution Consider a binomial experimentB(n, p). That is,B(n, p) consists ofnindependent re ...

136 PROBABILITY [CHAP. 7 Thus we can conclude that the probability that a value ofXlies between 65 and 85 is at least 1−( 1 / 2 ...

CHAP. 7] PROBABILITY 137 (a) Express explicitly the following events: A={heads and an even number}, B={prime number}, C={tails a ...

138 PROBABILITY [CHAP. 7 7.4.A single card is drawn from an ordinary deck of 52 cards. (See Fig. 7-2.) Find the probabilitypthat ...

CHAP. 7] PROBABILITY 139 FINITE PROBABILITY SPACES 7.9.A sample spaceSconsists of four elements; that is,S={a 1 ,a 2 ,a 3 ,a 4 } ...

140 PROBABILITY [CHAP. 7 Fig. 7-6 7.14. Prove Theorem 7.4 (Addition Principle): For any eventsAandB, P(A∪B)=P (A)+P(B)−P(A∩B) As ...

CHAP. 7] PROBABILITY 141 (d) Students who failed neither mathematics nor chemistry form the complement of the setM∪C, that is, t ...

142 PROBABILITY [CHAP. 7 (d) We have P(neitherAnorB)=P ((A∪B)C)= 1 −P(A∪B)= 1 − 157 = 158 7.20. Consider the following events fo ...

CHAP. 7] PROBABILITY 143 REPEATED TRIALS, BINOMIAL DISTRIBUTION 7.23. Suppose that, whenever horsesa,b,c,drace together, their r ...

144 PROBABILITY [CHAP. 7 7.27. Prove Theorem 7.7: The probability of exactlyksuccesses in a binomial experimentB(n, p)is given b ...