SAMPLE SPACES AND EVENTS
3.25. Let A and B be events. Find an expression and exhibit the Venn diagram for the event that (i) A or not B occurs, (ii) neither A nor B occurs.
3.26. Let A, B and C be events. Find an expression and exhibit the Venn diagram for the event that (i) exactly one of the three events occurs, (ii) at least two of the events occurs, (iii) none of the events occurs, (iv) A or B, but not C, occurs.
3.27. Let a penny, a dime and a die be tossed.
(i) Describe a suitable sample space S.
(ii) Express explicitly the following events: A = {two heads and a n even number appear}, B = {a 2 appears}, C = {exactly one head and a prime number appear}.
(iii) Express explicitly the event t h a t (a)A and B occur, (b) only B occurs, (c) B or C occurs.
FINITE PROBABILITY SPACES
3.28. Which function defines a probability space on
...
S = {al,a2,a,}'?(i) P(al)=
6,
P(a2)= 1, 8 P(a3 ) = 4 (111) P(a1)=Q,
P(a2)=a,
P(a,) =4
(ii) P(ai) =
3,
=-&,
P(a3)=+
(iv) p(al)= 0, ~ ( a , )=9,
~ ( a , )=8
3.29. Let P be a probability function on S = { a l , a 2 , a 3 } .Find P(al) if (i) P(a2) =
g
and P(a3)= $, (ii) P(al) = 2P(a2) and P(a3) =$,
(iii) P({a2,a,}) = 2P(a1), (iv) P(a,) = 2 P ( a 2 ) and P(a2)= 3P(a,).3.30. A coin is weighted so that heads is three times as likely to appear as tails. Find P(H)and P(T).
3.31. Three students A, B and C are in a swimming race. A and B have the same probability of winning and each is twice a s likely to win as C. Find the probability that B or C wins.
3.32. A die is weighted so that the even numbers have the same chance of appearing, the odd numbers have the same chance of appearing, and each even number is twice as likely to appear as any odd number. Find the probability that (i) an even number appears, (ii) a prime number appears, (iii) an odd number appears, (iv) a n odd prime number appears.
3.33. Find the probability of an event if the odds that i t will occur are (i) 2 to 1, (ii) 5 to 11.
3.34. In a swimming race, the odds that A will win are 2 to 3 and. the odds t h a t B will win are 1 to 4.
Find the probability p and the odds that A or B wins the race.
FINITE EQUIPROBABLE SPACES
3.35. A class contains 5 freshmen, 4 sophomores, 8 juniors and 3 seniors. A student is chosen at random to represent the class. Find the probability that the student is (i) a sophomore, (ii) a senior, (iii) a junior or senior.
3.36. One card is selected a t random from 50 cards numbered 1 to 50. Find the probability t h a t the number on the card is (i) divisible by 5, (ii) prime, (iii) ends in the digit 2.
3.37. Of 10 girls in a class, 3 have blue eyes. If two of the girls are chosen at random, what is the probability that (i) both have blue eyes, (ii) neither has blue eyes, (iii) at least one has blue eyes?
3.38. Three bolts and three nuts are put in a box. If two parts are chosen at random, find the probability that one is a bolt and one a nut.
52 INTRODUCTION TO PROBABILITY [CHAP. 3
3.39. Ten students, A , B ,
. . .,
are in a class. If a committee of 3 is chosen at random from the class, find the probability that (i) A belongs to the committee, (ii) B belongs to the committee,(iii) A and B belong to the committee, (iv) A or B belongs to the committee.
3.40. A class consists of 6 girls and 10 boys. If a committee of 3 is chosen at random from the class, find the probability that (i) 3 boys are selected, (ii) exactly 2 boys are selected, (iii) at least one boy is selected, (iv) exactly 2 girls are selected.
3.41. A pair of fair dice is tossed. Find the probability that the maximum of the two numbers is greater than 4.
3.42. Of 120 students, 60 are studying French, 60 are studying Spanish, and 20 are studying French and Spanish. If a student is chosen at random, find the probability that the student (i) is studying French or Spanish, (ii) is studying neither French nor Spanish.
3.43. Three boys and 3 girls sit in a row. Find the probability that (i) the 3 girls sit together, (ii) the boys and girls sit in alternate seats.
NONCOUNTABLE UNIFORM SPACES
3.44. A point is selected at random inside an equilateral triangle whose side length is 3. Find the probability that its distance to any corner is greater than 1.
3.45. A coin of diameter
-&
is tossed randomly onto the Cartesian plane R2. Find the probability that the coin does not intersect any line whose equation is of the form (a)x = k, (b) x+
= k, (c) x = kor y = k. (Here k is an integer.)
3.46. A point X is selected a t random from a line segment A B with midpoint 0. Find the probability that the line segments A X , XB and A 0 can form a triangle.
MISCELLANEOUS PROBLEMS
3.47. Let A and B be events with P ( A u B )=
8,
P ( A n B )=&
and P(Ac) =8.
Find P ( A ) ,P(B) and P ( AnBe).3.48. Let A and B be events with P ( A )=
8,
P ( A u B )=3
and P(Bc) =If.
Find P ( A n B ) , P(AcnBc), P(AcuBc) and P(BnAc).3.49. A die is tossed 50 times. The following table gives the six numbers and their frequency of occurrence:
Number
f i l
Find the relative frequency of the event (i) a 4 appears, (ii) an odd number appears, (iii) a prime number appears.
3.50. Prove: For any events A l , A,,
. . .,
A,,P ( A l U . * * U A , ) = X P ( A , )
-
i 7 j P ( A , n A j )+
i < y < k P ( A , n A j n A k )-
. . * 2 P ( A , n - n A , )2
(Remark: This result generalizes Theorem 3.5 and Corollary 3.6.)
53
CHAP. 31 INTRODUCTION TO PROBABILITY
Answers to Supplementary Problems
3.25. (i) AuBc, (ii) ( A u B ) ~
3.26. (i) (AnBcnCC)
u
(BnAcnCC)u
(CnAcnBC) (iii) ( A u B L J C ) ~ (ii) (AnB)u
(AnC)u (BnC) (iv) (A UB)n
Cc3.27. (i) S = (HH1, HH2, HH3, HH4, HH5, HH6, HT1, HT2, HT3, HT4, HT5, HT6, TH1, TH2, TH3, TH4, TH5, TH6, TT1, TT2, TT3, TT4, TT5, TT6)
(ii) A =(HH2, HH4, HH6}, B =(HH2, HT2, TH2, TT2), C = (HT2, TH2, HT3, TH3, HT5, TH5}
(iii) (a) A n B = (HH2) (b) B\(AuC) = (TT2)
(c) B u C = (HH2, HT2, TH2, TT2, HT3, TH3, HT5, TH5) 3.28. (i) no, (ii) no, (iii) yes, (iv) yes
3.29.
3.30. P ( H ) =
9,
P ( T ) =4
3.31.
6
3.32. (i)
Q ,
(ii)4,
(iii)9,
(iv)8
3.33.
3.34. p =
#;
the odds are 3 to 2.3.35.
3.36.
3.37.
3.38.
3.39.
3.40.
3.41.
3.42.
3.43.
3.44. 1
-
2 a / ( 9 6 ) 3.45.3.46.
3.47.
3.48.
3.49.