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mathematics
a first course in probability

Questions and Answers of A First Course In Probability

Two cards are randomly selected from an ordinary playing deck. What is the probability that they form a blackjack? That is, what is the probability that one of the cards is an ace and the other one
Let fn denote the number of ways of tossing a coin times such that successive heads never appear. Argue that fn = fn-1+fn-2 n2 2, where fo = 1, f, = 2 %3!
Two symmetric dice have had two of their sides painted red, two painted black, one painted yellow, and the other painted white. When this pair of dice is rolled, what is the probability that both
An urn contains red and m blue balls. They are withdrawn one at a time until a total of r, r ≤ n, red balls have been withdrawn. Find the probability that a total of balls are withdrawn.
Consider an experiment whose sample space consists of a countably infinite number of points. Show that not all points can be equally likely. Can all points have a positive probability of occurring?
If two dice are rolled, what is the probability that the sum of the upturned faces equals i? Find it for i = 2, 3, ... , 11, 12.
The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a 2, 3, or 12, the player loses; if the sum is either a 7 or an 11, the player wins. If the outcome
A pair of dice is rolled until a sum of either 5 or 7 appears. Find the probability that a 5 occurs first.
An urn contains n white and m black balls, where and m are positive numbers.a. If two balls are randomly withdrawn, what is the probability that they are the same color?b. If a ball is randomly
A 3-person basketball team consists of a guard, a forward, and a center.a. If a person is chosen at random from each of three different such teams, what is the probability of selecting a complete
A group of individuals containing boys and girls is lined up in random order; that is, each of the (b + g)! permutations is assumed to be equally likely. What is the probability that the person in
A forest contains 20 elk, of which 5 are captured, tagged, and then released. A certain time later, 4 of the 20 elk are captured. What is the probability that 2 of these 4 have been tagged? What
The second Earl of Yarborough is reported to have bet at odds of 1000 to 1 that a bridge hand of 13 cards would contain at least one card that is ten or higher. (By ten or higher we mean that a card
There are n socks, 3 of which are red, in a drawer. What is the value of n if, when 2 of the socks are chosen randomly, the probability that they are both red is 1/2?
There are 5 hotels in a certain town. If 3 people check into hotels in a day, what is the probability that they each check into a different hotel? What assumptions are you making?
If a die is rolled 4 times, what is the probability that 6 comes up at least once?
Two dice are thrown n times in succession. Compute the probability that double 6 appears at least once. How large need n be to make this probability at least 1/2?
A woman has keys, of which one will open her door.a. If she tries the keys at random, discarding those that do not work, what is the probability that she will open the door on her kth try?b. What if
Suppose that of the numbers 1, 2, ... , 14 are chosen. Find the probability that 9 is the third smallest value chosen.
Given 20 people, what is the probability that among the 12 months in the year, there are 4 months containing exactly 2 birthdays and 4 containing exactly 3 birthdays?
In a hand of bridge, find the probability that you have 5 spades and your partner has the remaining 8.
A group of 6 men and 6 women is randomly divided into 2 groups of size 6 each. What is the probability that both groups will have the same number of men?
If 8 people, consisting of couples, are randomly arranged in a row, find the probability that no person is next to their partner.
Show that if P(A) > 0, thenP(AB|A) ≥ P(AB|A ∪ B)
What is the probability that at least one of a pair of fair dice lands on 6, given that the sum of the dice is i, i = 2, 3, ..., 12?
a. Prove that if E and F are mutually exclusive, thenb. Prove that Ei, i ≥ 1 if are mutually exclusive, then P(E) P(E) + P(F) P(E\E U F) =
An urn contains 6 white and 9 black balls. If 4 balls are to be randomly selected without replacement, what is the probability that the first 2 selected are white and the last 2 black?
The king comes from a family of 2 children. What is the probability that the other child is his sister?
A couple has 2 children. What is the probability that both are girls if the older of the two is a girl?
Two percent of women age 45 who participate in routine screening have breast cancer. Ninety percent of those with breast cancer have positive mammography's. Eight percent of the women who do not have
Three cards are randomly selected, without replacement, from an ordinary deck of 52 playing cards. Compute the conditional probability that the first card selected is a spade given that the second
In each of n independent tosses of a coin, the coin lands on heads with probability p. How large need be so that the probability of obtaining at least one head is at least 1/2?
Show that 0 ≤ ai ≤ 1, i = 1, 2, ...., thenSuppose that an infinite number of coins are to be flipped. Let ai be the probability that the ith coin lands on heads, and consider when the first
Suppose distinct values are written on each of cards, which are then randomly given the designations A, B and C. Given that card A’s value is less than card B’s value, find the probability it is
Suppose that you are gambling against an infinitely rich adversary and at each stage you either win or lose 1 unit with respective probabilities and a - p. Show that the probability that you
An urn initially contains 5 white and 7 black balls. Each time a ball is selected, its color is noted and it is replaced in the urn along with 2 other balls of the same color. Compute the probability
Suppose that independent trials are performed, with trial being a success with probability 1/(2i + 1). Let Pn denote the probability that the total number of successes that result is an odd number.a.
Let Qn denote the probability that no run of 3 consecutive heads appears in n tosses of a fair coin. Show thatFind Q8. 1 1 +-Qn-2 +3 en-3 1 2 n-1 Q1 = Q2 = 1
Consider the gambler’s ruin problem, with the exception that A and B agree to play no more than games. Let Pn,i denote the probability that A winds up with all the money when starts with i and B
A bag contains white and black balls. Balls are chosen from the bag according to the following method:A. A ball is chosen at random and is discarded.B. A second ball is then chosen. If its color is
Twenty percent of B’s phone calls are with her daughter. Sixty five percent of the time that B speaks with her daughter she hangs up the phone with a smile on her face. Given that B has just hung
The following method was proposed to estimate the number of people over the age of 50 who reside in a town of known population 100,000: “As you walk along the streets, keep a running count of the
Prove or give a counterexample. If E1 and E2 are independent, then they are conditionally independent given F.
Prove the generalized version of the basic counting principle.
Two experiments are to be performed. The first can result in any one of m possible outcomes. If the first experiment results in outcome then the second experiment can result in any of ni possible
How many outcome sequences are possible when a die is rolled four times, where we say, for instance, that the outcome is 3, 4, 3, 1 if the first roll landed on 3, the second on 4, the third on 3, and
In how many ways can r objects be selected from a set of n objects if the order of selection is considered relevant?
Thereare different linear arrangements of n balls of which r are black and n - r are white. Give a combinatorial explanation of this fact.
John, Jim, Jay, and Jack have formed a band consisting of 4 instruments. If each of the boys can play all 4 instruments, how many different arrangements are possible? What if John and Jim can play
Prove that1. (*") - X)-Q(-) ()) (n+m m m +... +
For years, telephone area codes in the United States and Canada consisted of a sequence of three digits. The first digit was an integer between 2 and 9, the second digit was either 0 or 1, and the
How many vectors x1, ... xk, are there for which each xi is a positive integer such that 1 ≤ xi ≤ ni and xi < x2 < ··· < xk?
A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is blue. If the child puts the blocks in a line, how many arrangements are possible?
In how many ways can 3 novels, 2 mathematics books, and 1 chemistry book be arranged on a bookshelf ifa. The books can be arranged in any order?b. The mathematics books must be together and the
How many digit numbers xyz, with x, y, z, all ranging from to have at least of their digits equal. How many have exactly equal digits.
Show that, for n > 0, n = 0 i = 0
How many different letter permutations, of any length, can be made using the letters M O T T O. (For instance, there are possible permutations of length )
Five separate awards (best scholarship, best leadership qualities, and so on) are to be presented to selected students from a class of 30. How many different outcomes are possible ifa. a student can
Consider a group of 20 people. If everyone shakes hands with everyone else, how many handshakes take place?
How many 5-card poker hands are there?
Argue that1. n п — 1 n1,n2, . . .,nr n1 - 1, n2, . . .n, п - 1 +. n1, n2 - 1, . . .,n, п - 1 n1, n2,. .,nr - 1
Seven different gifts are to be distributed among 10 children. How many distinct results are possible if no child is to receive more than one gift?
A committee of 7, consisting of 2 Republicans, 2 Democrats, and 3 Independents, is to be chosen from a group of 5 Republicans, 6 Democrats, and 4 Independents. How many committees are possible?
Consider a function f(x1, ..., xn) of n variables. How many different partial derivatives of order r does f possess?
A person has 8 friends, of whom 5 will be invited to a party.a. How many choices are there if 2 of the friends are feuding and will not attend together?b. How many choices if 2 of the friends will
A psychology laboratory conducting dream research contains 3 rooms, with 2 beds in each room. If 3 sets of identical twins are to be assigned to these 6 beds so that each set of twins sleeps in
a. Showb. Simplify n |2* = 3" k k=0
Expand (3x2 + y)5.
The game of bridge is played by 4 players, each of whom is dealt 13 cards. How many bridge deals are possible?
Expand (x1 + 2x2 + 3x3)4.
If 12 people are to be divided into 3 committees of respective sizes 3, 4, and 5, how many divisions are possible?
If 8 new teachers are to be divided among 4 schools, how many divisions are possible? What if each school must receive 2 teachers?
Ten weight lifters are competing in a team weight-lifting contest. Of the lifters, 3 are from the United States, 4 are from Russia, 2 are from China, and 1 is from Canada. If the scoring takes
If 8 identical blackboards are to be divided among 4 schools, how many divisions are possible? How many if each school must receive at least 1 blackboard?
An elevator starts at the basement with 8 people (not including the elevator operator) and discharges them all by the time it reaches the top floor, number 6. In how many ways could the operator have
Prove the following relations:EF ⊂ E ⊂ E ∪ F.
Prove the following relations:If E ⊂ F, then Fc ⊂ Ec.
Prove the following relations:F = FE, ∪ FEc and E ∪ F = E ∪ EcF.
A, B, and C take turns flipping a coin. The first one to get a head wins. The sample space of this experiment can be defined bya. Interpret the sample space.b. Define the following events in terms of
For any sequence of events E1, E2,..., define a new sequence F1, F2,... of disjoint events (that is, events such that FiFj = Ø whenever i ≠ j) such that for all n ≥ 1, n UF; = UE, U E; 1,
Let E, F and be three events. Find expressions for the events so that, of E, F and G,a. Only occurs;b. Both E and G, but not F, occur;c. At least one of the events occurs;d. At least two of the
Consider an experiment that consists of determining the type of job–either blue collar or white collar–and the political affiliation Republican, Democratic, or Independent–of the 15 members of
Use Venn diagramsa. To simplify the expression (E ∪ F) (E ∪ Fc);b. To prove DeMorgan’s laws for events E and F. [That is, prove (EUF) = E°F°, and (EF) = E°UF]
Suppose that and are mutually exclusive events for which P(A) = .3. and P(B) = .5. What is the probability thata. Either or occurs?b. Occurs but does not?c. Both and occur?
Suppose that an experiment is performed times. For any event of the sample space, let n(E) denote the number of times that event occurs and define f(E) = n(E)/n. Show that f(·) satisfies Axioms 1,
If P(E) = .9 and P(F) = .8 show that P(EF) ≥ .7. In general, proveBonferroni’s inequality, namely,P(EF) ≥ P(E) + P(F) = -1
Show that the probability that exactly one of the events or occursequals P(E) + P(F) - 2P(EF).
Prove thatP(EFc) = P(E) - P(EF).
An urn contains M white and black balls. If a random sample of size is chosen, what is the probability that it contains exactly k white balls?
Twenty five people, consisting of 15 women and 10 men are lined up in a random order. Find the probability that the ninth woman to appear is in position 17. That is, find the probability there are
Consider the matching problem, Example 5m, and define AN to be the number of ways in which the men can select their hats so that no man selects his own. Argue thatThis formula, along with the

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