Decoding Probabilities: Unions, Intersections, and Complements

PROBLEM 16E Let pn, n = 0, 1, 2, . . . , be the probability that an automobile policyholder will file for n claims in a five-year period. The actuary involved makes the assumption that pn+1 = (1/4)pn. What is the probability that the holder will file two or more claims during this period?

PROBLEM 2E An insurance company looks at its auto insurance customers and finds that (a) all insure at least one car, (b) 85% insure more than one car, (c) 23% insure a sports car, and (d) 17% insure more than one car, including a sports car. Find the probability that a customer selected at random insures exactly one car and it is not a sports car.

PROBLEM 1E Of a group of patients having injuries, 28% visit both a physical therapist and a chiropractor and 8% visit neither. Say that the probability of visiting a physical therapist exceeds the probability of visiting a chiropractor by 16%. What is the probability of a randomly selected person from this group visiting a physical therapist?

Draw one card at random from a standard deck of cards. The sample space \(S\) is the collection of the 52 cards. Assume that the probability set function assigns \(1 / 52\) to each of the 52 outcomes. Let \(A = {x: x\) is a jack, queen, or king}, \(B = {x: x\) is a 9, 10, or jack and x is red}, \(C = {x: x\) is a club}, \(D = {x: x\) is a diamond, a heart, or a spade}. Find (a) \(P(A)\), (b) \(P(A \cap B)\), (c) \(P(A \cup B)\), (d) \(P(C \cup D)\), and (e) \(P(C \cap D)\). Equation Transcription: Text Transcription: S 1/52 A={x:x B={x:x C={x:x D={x:x P(A) P(A cap B) P(A cup B) P(C cup D) P(C cap D)

A fair coin is tossed four times, and the sequence of heads and tails is observed. (a) List each of the 16 sequences in the sample space S. (b) Let events A, B, C, and D be given by A = {at least 3 heads}, B = {at most 2 heads}, C = {heads on the third toss}, and D = {1 head and 3 tails}. If the probability set function assigns 1/16 to each outcome in the sample space, find (i) P(A), (ii) \(P(A \cap B)\), (iii) P(B), (iv) \(P(A \cap C)\), (v) P(D), (vi) \(P(A \cup C)\), and (vii) \(P(B \cap D)\).

Consider the trial on which a 3 is first observed in successive rolls of a six-sided die. Let A be the event that 3 is observed on the first trial. Let B be the event that at least two trials are required to observe a 3. Assuming that each side has probability 1/6, find (a) P(A), (b) P(B), and (c) P(A ? B).

PROBLEM 7E Given that P(A ? B) = 0.76 and P(A ? B' ) = 0.87, find P(A).

PROBLEM 6E If P(A) = 0.4, P(B) = 0.5, and P(A?B) = 0.3, find (a) P(A ? B), (b) P(A ? B' ), and (c) P(A' ? B' ).

PROBLEM 8E During a visit to a primary care physician’s office, the probability of having neither lab work nor referral to a specialist is 0.21. Of those coming to that office, the probability of having lab work is 0.41 and the probability of having a referral is 0.53.What is the probability of having both lab work and a referral?

PROBLEM 11E A typical roulette wheel used in a casino has 38 slots that are numbered 1, 2, 3, . . . , 36, 0, 00, respectively. The 0 and 00 slots are colored green. Half of the remaining slots are red and half are black. Also, half of the integers between 1 and 36 inclusive are odd, half are even, and 0 and 00 are defined to be neither odd nor even. A ball is rolled around the wheel and ends up in one of the slots; we assume that each slot has equal probability of 1/38, and we are interested in the number of the slot into which the ball falls. (a) Define the sample space S. (b) Let A = {0, 00}. Give the value of P (A). (c) Let B = {14, 15, 17, 18}. Give the value of P (B). (d) Let D = {x : x is odd}. Give the value of P (D).

PROBLEM 12E Let x equal a number that is selected randomly from the closed interval from zero to one, [0, 1]. Use your intuition to assign values to (a) P({x: 0 ? x ? 1/3}). (b) P({x: 1/3 ? x ? 1}). (c) P({x: x = 1/3}). (d) P({x: 1/2 < x < 5}).

PROBLEM 13E Divide a line segment into two parts by selecting a point at random. Use your intuition to assign a probability to the event that the longer segment is at least two times longer than the shorter segment.

Roll a fair six-sided die three times. Let \(A_{1}=\) \(\{1\) or 2 on the first roll \(\}, A_{2}=\{3\) or 4 on the second roll \(\}\), and \(A_{3}=\{5\) or 6 on the third roll \(\}\). It is given that \(P\left(A_{i}\right)=1 / 3, i=1,2,3 ; P\left(A_{i} \cap A_{j}\right)=(1 / 3)^{2}, i \neq j ;\) and \(P\left(A_{1} \cap A_{2} \cap A_{3}\right)=(1 / 3)^{3}\). (a) Use Theorem 1.1-6 to find \(P\left(A_{1} \cup A_{2} \cup A_{3}\right)\). (b) Show that \(P\left(A_{1} \cup A_{2} \cup A_{3}\right)=1-(1-1 / 3)^{3}\). Equation Transcription: Text Transcription: A_1= {1 },A_2={3 A_3={5 P(A_i)=1/3, i=1,2,3;P(A_i cap A_j)=(1/3)^2, i not = j; P(A_1 cap A_2 cap A_3)=(1/3)^3 P(A_1 cup A_2 cup A_3) P(A_1 cup A_2 cup A_3)=1-(1-1/3)^3

PROBLEM 14E Let the interval [?r, r] be the base of a semicircle. If a point is selected at random from this interval, assign a probability to the event that the length of the perpendicular segment from the point to the semicircle is less than r/2.

PROBLEM 15E Let S = A1 ? A2 ? · · · ? Am, where events A1,A2, . . . ,Am are mutually exclusive and exhaustive. (a) If P(A1) = P(A2) = · · · = P(Am), show that P(Ai) = 1/m, i = 1, 2, . . . ,m. (b) If A = A1 ?A2?· · ·?Ah, where h < m, and (a) holds, prove that P(A) = h/m.

Prove Theorem 1.1-6.

Of a group of patients having injuries, 28% visit both a physical therapist and a chiropractor and 8% visit neither. Say that the probability of visiting a physical therapist exceeds the probability of visiting a chiropractor by 16%. What is the probability of a randomly selected person from this group visiting a physical therapist?

An insurance company looks at its auto insurance customers and finds that (a) all insure at least one car, (b) 85% insure more than one car, (c) 23% insure a sports car, and (d) 17% insure more than one car, including a sports car. Find the probability that a customer selected at random insures exactly one car and it is not a sports car.

Draw one card at random from a standard deck of cards. The sample space S is the collection of the 52 cards. Assume that the probability set function assigns 1/52 to each of the 52 outcomes. Let A = {x: x is a jack, queen, or king}, B = {x: x is a 9, 10, or jack and x is red}, C = {x: x is a club}, D = {x: x is a diamond, a heart, or a spade}. Find (a) P(A), (b) P(A B), (c) P(A B), (d) P(C D), and (e) P(C D).

A fair coin is tossed four times, and the sequence of heads and tails is observed. (a) List each of the 16 sequences in the sample space S. (b) Let events A, B, C, and D be given by A = {at least 3 heads}, B = {at most 2 heads}, C = {heads on the third toss}, and D = {1 head and 3 tails}. If the probability set function assigns 1/16 to each outcome in the sample space, find (i) P(A), (ii) P(A B), (iii) P(B), (iv) P(A C), (v) P(D), (vi) P(A C), and (vii) P(B D).

Consider the trial on which a 3 is first observed in successive rolls of a six-sided die. Let A be the event that 3 is observed on the first trial. Let B be the event that at least two trials are required to observe a 3. Assuming that each side has probability 1/6, find (a) P(A), (b) P(B), and (c) P(A B).

If P(A) = 0.4, P(B) = 0.5, and P(AB) = 0.3, find (a) P(A B), (b) P(A B ), and (c) P(A B ).

Given that \(P(A \cup B) = 0.76\) and \(P(A \cup B ) = 0.87\), find P(A).

During a visit to a primary care physicians office, the probability of having neither lab work nor referral to a specialist is 0.21. Of those coming to that office, the probability of having lab work is 0.41 and the probability of having a referral is 0.53. What is the probability of having both lab work and a referral?

Roll a fair six-sided die three times. Let A1 = {1 or 2 on the first roll}, A2 = {3 or 4 on the second roll}, and A3 = {5 or 6 on the third roll}. It is given that P(Ai) = 1/3, i = 1, 2, 3; P(Ai Aj) = (1/3)2, i = j; and P(A1 A2 A3) = (1/3)3. (a) Use Theorem 1.1-6 to find P(A1 A2 A3). (b) Show that P(A1 A2 A3) = 1 (1 1/3)3.

Prove Theorem 1.1-6.

A typical roulette wheel used in a casino has 38 slots that are numbered 1, 2, 3, ... , 36, 0, 00, respectively. The 0 and 00 slots are colored green. Half of the remaining slots are red and half are black. Also, half of the integers between 1 and 36 inclusive are odd, half are even, and 0 and 00 are defined to be neither odd nor even. A ball is rolled around the wheel and ends up in one of the slots; we assume that each slot has equal probability of 1/38, and we are interested in the number of the slot into which the ball falls. (a) Define the sample space S. (b) Let A = {0, 00}. Give the value of P(A). (c) Let B = {14, 15, 17, 18}. Give the value of P(B). (d) Let D = {x : x is odd}. Give the value of P(D).

Let x equal a number that is selected randomly from the closed interval from zero to one, [0, 1]. Use your intuition to assign values to (a) P({x: 0 x 1/3}). (b) P({x: 1/3 x 1}). (c) P({x: x = 1/3}). (d) P({x: 1/2 < x < 5}).

Divide a line segment into two parts by selecting a point at random. Use your intuition to assign a probability to the event that the longer segment is at least two times longer than the shorter segment.

Let the interval [r,r] be the base of a semicircle. If a point is selected at random from this interval, assign a probability to the event that the length of the perpendicular segment from the point to the semicircle is less than r/2.

Let S = A1 A2 Am, where events A1, A2, ... , Am are mutually exclusive and exhaustive. (a) If P(A1) = P(A2) = = P(Am), show that P(Ai) = 1/m, i = 1, 2, ... , m. (b) If A = A1 A2 Ah, where h < m, and (a) holds, prove that P(A) = h/m.

Let pn, n = 0, 1, 2, ... , be the probability that an automobile policyholder will file for n claims in a five-year period. The actuary involved makes the assumption that pn+1 = (1/4)pn. What is the probability that the holder will file two or more claims during this period?