4.1a 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 1E4: Observe a 4

E2: Observe a 2E5: Observe a 5

E3: Observe a 3E6: Observe a 6

bEvents 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

cSince the simple events Ei, i = 1, 2, 3, …, 6 are equally likely, [pic]. dTo 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].

4.2aIt 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]

bTo find the necessary probabilities, sum the probabilities of the simple events:

[pic]

c-dThe following events are in either A or B or both: { E1, E2, E3, E4}. Only event E3 is in both A and B.

4.3It 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]

4.4aIt is required that [pic]. Hence, [pic]

bThe 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]

4.5aThe experiment consists of choosing three coins at random from four. The order in which the coins are drawn is unimportant. Hence, each simple event consists of a triplet, indicating the three coins drawn. Using the letters N, D, Q, and H to represent the nickel, dime, quarter, and half-dollar, respectively, the four possible simple events are listed below. E1: (NDQ)E2: (NDH)E3: (NQH)E4: (DQH)

bThe event that a half-dollar is chosen is associated with the simple events E2, E3, and E4. Hence, [pic]

since each simple event is equally likely.

cThe simple events along with their monetary values follow:

E1 NDQ $0.40

E2 NDH 0.65

E3 NQH 0.80

E4 DQH 0.85

Hence, [pic].

4.6aThe experiment consists of selecting one of 25 students and recording the student’s gender as well as whether or not the student had gone to preschool.

bThe experiment is accomplished in two stages, as shown in the tree diagram below.

GenderPreschoolSimple EventsProbability

YesE1 : Male, Preschool 8/25

Male

NoE2 : Male, No preschool 6/25

YesE3 : Female, Preschool 9/25

Female

NoE4 : Female, No preschool 2/25

cSince each of the 25 students are equally likely to be chosen, the probabilities will be proportional to the number of students in each of the four gender-preschool categories. These probabilities are shown in the last column of the tree diagram above. d[pic]

4.7Label the five balls as R1, R2, R3, Y1 and Y2. The selection of two balls is accomplished in two stages to produce the simple events in the tree diagram on the next page.

First BallSecond BallSimple EventsFirst Ball Second Ball Simple Events R2R1R2R1 Y1R1

R3R1R3R2 Y1R2

R1 Y1R1Y1 Y1R3 Y1R3

Y2R1Y2Y2 Y1Y2

R1R2R1R1 Y2R1

R2 R3R2R3...