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

150 PROBABILITY [CHAP. 7

7.54. Consider the following probability distribution:

Outcomex 12345

ProbabilityP(x) 0.2 0.4 0.1 0.1 0.2

Consider the eventsA={even number},B={ 2 , 3 , 4 , 5 },C={ 1 , 2 }. Find:

(a)P (A),P(B),P(C); (b)P(A∩B),P(A∩C),P(B∩C).

7.55. SupposeAandBare events withP (A)= 0 .7,P(B)= 0 .5, andP(A∩B)= 0 .4. Find the probability that:

(a)Adoes not occur; (c)Abut notBoccurs;

(b)AorBoccurs; (d) neitherAnorBoccurs.

CONDITIONAL PROBABILITY, INDEPENDENCE

7.56.A fair die is tossed. Consider eventsA={ 2 , 4 , 6 },B={ 1 , 2 },C={ 1 , 2 , 3 , 4 }. Find:

(a)P(AandB) andP(AorC), (c)P(A|C)andP(C|A)

(b)P(A|B)andP(B|A) (d)P(B|C)andP(C|B)

Decide whether the following are independent: (i)AandB; (ii)AandC; (iii)Band C.

7.57.Apair of fair dice is tossed. If the numbers appearing are different, find the probability that: (a) the sum is even; (b) the

sum exceeds nine.

7.58. LetAandBbe events withP (A)= 0 .6,P(B)= 0.3, andP(A∩B)= 0 .2. Find:

(a)P(A∪B); (b)P(A|B); (c)P(B|A).

7.59. LetAandBbe events withP (A)= 1 /3,P(B)=^14 , andP(A∪B)=^12.

(a) FindP(A|B)andP(B|A). (b) AreAandBindependent?

7.60. LetAandBbe events withP (A)= 0 .3,P(A∪B)= 0 .5, andP(B)=p. Findpif:

(a)AandBare mutually disjoint; (b)AandBare independent; (c)Ais a subset ofB.

7.61. LetAandBbe independent events withP (A)= 0 .3 andP(B)= 0 .4. Find:

(a)P(A∩B)andP(A∪B); (b)P(A|B)andP(B|A).

7.62. In a country club, 60% of the women play tennis, 40% play golf, and 20% play both tennis and golf. A woman is

chosen at random.

(a) Find the probability that she plays neither tennis nor golf.

(b) If she plays tennis, find the probability that she plays golf.

(c) If she plays golf, find the probability that she plays tennis.

7.63. BoxAcontains six red marbles and two blue marbles, and boxBcontains two red and four blue. A marble is drawn

at random from each box.

(a) Find the probabilitypthat both marbles are red.

(b) Find the probabilitypthatone is red and one is blue.

7.64. Theprobability thatAhits a target is^14 and the probability thatBhits a target is^13.

(a) If each fires twice, what is the probability that the target will be hit at least once?

(b) If each fires once and the target is hit only once, what is the probability thatAhits the target?

7.65. Three fair coins are tossed. Consider the events:

A={all heads or all tails}, B={at least two heads}, C={at most two heads}.

Of the pairs(A, B),(A, C), and(B, C), which are independent? Which are dependent?

7.66. FindP(B|A)if: (a)Ais a subset ofB; (b)AandBare mutually exclusive. (AssumeP (A) >0.)