Answer Chapter 4 Introduction to Statistics and Probability Edition 13th
4: Probability and Probability Distributions
4.1 a This experiment involves tossing a single die and observing the outcome. The sample space for this experiment consists of the following simple events: E1: Observe a 1 E4: Observe a 4 E2: Observe a 2 E5: Observe a 5 E3: Observe a 3 E6: Observe a 6
b Events A through F are compound events and are composed in the following manner: A: (E2) D: (E 2) B: (E 2, E 4, E 6) E: (E 2, E 4, E6) C: (E 3, E 4, E 5, E 6) F: contains no simple events
c Since the simple events Ei, i = 1, 2, 3, …, 6 are equally likely, [pic]. d To find the probability of an event, we sum the probabilities assigned to the simple events in that event. For example, [pic] Similarly, [pic] and [pic] Since event F contains no simple events, [pic].
4.2 a It is given that [pic] and [pic]. Since [pic], we know that [pic] (i)
Also, it is given that [pic] (ii) We have two equations in two unknowns which can be solved simultaneously for P(E 4) and P(E 5). Substituting equation (ii) into equation (i), we have [pic]
b To find the necessary probabilities, sum the probabilities of the simple events:
[pic]
c-d The following events are in either A or B or both: { E1, E2, E3, E4}. Only event E3 is in both A and B.
4.3 It is given that [pic]and that [pic], so that [pic]. Since [pic], the remaining 8 simple events must have probabilities whose sum is [pic]. Since it is given that they are equiprobable, [pic]
4.4 a It is required that [pic]. Hence, [pic] b The player will hit on at least one of the two freethrows if he hits on the first, the second, or both. The associated simple events are E1, E2, and E3 and
[pic]
4.5 a The experiment consists of choosing three coins at random from four. The order in which the coins are drawn is
4.1 a This experiment involves tossing a single die and observing the outcome. The sample space for this experiment consists of the following simple events: E1: Observe a 1 E4: Observe a 4 E2: Observe a 2 E5: Observe a 5 E3: Observe a 3 E6: Observe a 6
b Events A through F are compound events and are composed in the following manner: A: (E2) D: (E 2) B: (E 2, E 4, E 6) E: (E 2, E 4, E6) C: (E 3, E 4, E 5, E 6) F: contains no simple events
c Since the simple events Ei, i = 1, 2, 3, …, 6 are equally likely, [pic]. d To find the probability of an event, we sum the probabilities assigned to the simple events in that event. For example, [pic] Similarly, [pic] and [pic] Since event F contains no simple events, [pic].
4.2 a It is given that [pic] and [pic]. Since [pic], we know that [pic] (i)
Also, it is given that [pic] (ii) We have two equations in two unknowns which can be solved simultaneously for P(E 4) and P(E 5). Substituting equation (ii) into equation (i), we have [pic]
b To find the necessary probabilities, sum the probabilities of the simple events:
[pic]
c-d The following events are in either A or B or both: { E1, E2, E3, E4}. Only event E3 is in both A and B.
4.3 It is given that [pic]and that [pic], so that [pic]. Since [pic], the remaining 8 simple events must have probabilities whose sum is [pic]. Since it is given that they are equiprobable, [pic]
4.4 a It is required that [pic]. Hence, [pic] b The player will hit on at least one of the two freethrows if he hits on the first, the second, or both. The associated simple events are E1, E2, and E3 and
[pic]
4.5 a The experiment consists of choosing three coins at random from four. The order in which the coins are drawn is