Questions and Answers of Probability And Stochastic Modeling

(a) Just looking at the table of joint probabilities and providing all calculations in mind, find P(X1 = 3|X2 = 0).(b) Just looking at the table of joint probabilities and providing all calculations
A husband and wife played the same slot machine together. The wife started to play first, played until the first win and yielded her place to the husband. Then the husband played until his first win
(a) Two independent r.v.’s, X1 and X2, take on values 1,2 with equal probabilities. Without calculations, guesswhether all possible values of the sum X1 + X2 are also equally likely. Give a
Find the distribution of the sum X1+X2 of independent r.v.’s
Regarding the convolution formula (1.2.5), is it true that f(1) ∗ f(2) = f(2) ∗ f(1)?
A husband and wife have two cars. They can purchase special auto insurance policies for each car separately, such that each policy covers the loss only for the first accident occurred in the same
Consider the game from Example 1.1-7, and suppose that the coin was tossed n times. Let Wn be the total winnings of the first player.(a) Which values can Wn assume?(b) Find P(Wn = 0), (c) Find P(Wn
Show without any calculations and using no particular formulas that the sum of two independent binomial r.v.’s with parameters (n1, p) and (n2, p) respectively, is a binomial r.v. with parameters
Let r.v.’s X1 and X2 assume the same values, and their joint distribution is symmetric in the sense that fij = f ji.(a) How will the table of joint probabilities look in this case?(b) Do X1,X2
Let r.v.’s X1 and X2 be i.i.d. Show that P(X1 > X2) = P(X2 > X1).
Are r.v.’s X1 ≡ 3 and X2 ≡ 5 independent? (X ≡ a means that X takes on only one value a.)
The two tables below give two different joint probability distributions of a random vector (X1,X2):For both cases, find the marginal probabilities; figure out whether the random variables are
Let r.v.’s X1,X2 be the numbers on two dice rolled. How does the table of joint and marginal probabilities look in this case? Realize how this simple table may be of help for finding the
Let r.v.’s X1 and X2 be independent and both have the geometric distribution with parameter p = 1/2. Find P(X1 = X2).
The distribution of the number of telephone calls a manager is receiving during an hour is well approximated by the Poisson distribution with parameter λ = 4. Calculate the probability that during
Let a r.v.Graph the (cumulative) distribution function of X. If you show this graph to somebody who is taking a course in Probability, how would she/he compute P(−3 ≤ X ≤ 3) just looking at the
There are six items in a row. You mark at random two of them. Find the distribution of the number of items between the marked items.
A bag contains 5 red marbles and 15 blue marbles. Two marbles are drawn from the bag without replacement. Write the distribution of the number of red marbles selected.
Let X be the product of the numbers on two dice rolled. (a) Does X assume all values between 1 and 36? (b) Trying to provide all calculations in mind (they are easy), find P(X ≤ 36), P(X < 36),
You draw two cards from a deck. If they are of different color, you win nothing. If both are black, you win $5. If both are red, you win $10, and additional $15 if the two cards are a red ace and
(a) Which distribution below would you call symmetric, and which “skewed”?(b) Rigorously speaking, the distribution of a discrete r.v. X is said to be symmetric if for a number s called a center
“Hickory, dickory, dock,The mouse ran up the clock.The clock struck one,The mouse ran down,Hickory, dickory, dock.”Suppose a word is chosen at random from this famous nursery rhyme. Write the
Suppose that in the situation of the two-armed bandit problem in Section 2.7, you follow the strategy suggested there.(a) Show that (i) if you lose at the first time, then you will switch; (ii) if
A company is advertising a product it produces. The management places an advertisement in newspapers, and a fancy advertisement on TV. The following average figures have been observed. Among 50
Using software, compute p3(4), p2(4), p3(10), p5(10), and p8(10) in the model of Section 3.6. Compare the results with what approximation (3.6.4) gives. Explain the result of the comparison, and
Ten guests came to a diner party. The host had had plans where each guest would sit, but the guests ignored it and chose seats at random. Find the probability that at most two guests chose the seats
Consider the matching scheme of Section 3.6. Explain without calculations why pn−1(n) must be equal to zero, and whymust be equal to one. Show that both assertions are consistent with (3.6.1).
Prove thatrigorously, and give a combinatorial interpretation. Proceeding from (4.1), show that all combinations (n k) can be arranged in the triangle below, which is constructed in the following
Using the binomial formula, prove that
(a) Which is larger:(b) For a fixed n, consider (n k) as a function of k. Figure out for which k it is increasing and for which it is decreasing. Where does it attain its maximal value for an even
Explain without any calculations why
There are 4 roses and 5 lilies. Five flowers are randomly selected. Find the probability that the bouquet will contain 2 roses and 3 lilies.
Ten women and nine men attend a lecture. Suppose that all orders in which they can leave the room after the lecture is over are equally likely. Find the probability that(a) All women will leave
Fifteen people are randomly seated in a row of thirty chairs. Write a formula for the probability that the people will occupy fifteen adjacent seats?
In a country, there are only five first names for boys, and all five names are equally likely to be given.(a) Find the probability that four boys selected at random have different names. (b) Find
(a) Write a counterpart of (2.5.4) for three events. (Advice: Start with P(A ∪ B ∪ C)= P((A ∪ B) ∪ C), and use (2.5.4).) (b) Prove (2.5.6) by induction. P(AUB)=P(A) + P(B) P(AB). (2.5.4)
Let P(A) = 0.9 and P(B) = 0.2. Can P(AB) = 0? In general, if P(A) + P(B) > 1, can P(AB) = 0? Eventually, show that P(AB) ≥ P(A) + P(B) − 1, and find to which value P(AB) can be equal if P(A) =
Let A and B be events, and let C be the event that exactly one of the events A or B occurs. Using operations on events, write a formula for C. Write a formula for P(C) in terms of P(A), P(B), and
In a city of Pleasant Corner, 20% of households have a pool, 60% have air conditioning, and 70% have at least one of these features. Find the probabilities of all possible combinations of these
In an area, for two consecutive days, the probability that it is raining on both is 0.1, and the probability that there is no rain on any of these two days is 0.6. Also, it is equally likely whether
Explain why it is impossible to build a discrete model with an infinite number of outcomes, for which all outcomes are equally likely.
Similar to Proposition 1, show that for any events A1, A2, ...,where ∪iAi and ∩iAi stand for the union and intersection of all Ai’s, respectively. (UA) = NA, (NA) = UA,
Mark each statement below true or false.(a) In any probability model, the number of all possible outcomes is finite.(b) In any probability model, all outcomes are equally likely.(c) P(A∪B) =
Prove the second relation in (1.2.2) replacing A by Ac and B by Bc in the first. (AUB) = AnB, (ANB) = AUB. (1.2.2)
In Example 1.2-4, describe AcBc.EXAMPLE 4. Let A={(x,y) ∈ Ω : x ≥ 0} and B = {(x,y) ∈ Ω : y ≥ 0}. The situation is illustrated in Fig. 3. The set A corresponds to the right half of the
Describe the sample space for the situations below and compute |Ω|.(a) There are two political parties in a country. Each of n citizens either votes for a party, or does not vote at all, or uses his
Two dice are rolled. Let A = {the sum of the dice is odd}, B = {at least one die is even}, C = {at least one die is odd}. For each pair of these events, figure out whether the events in the pair are
Suppose that A and B are disjoint, P (A) = 0.4 and P (B) = 0.5. Find the probabilities that (a) Either A or B occur;(b) Both A and B occur;(c) A occurs but B does not.
A point (a,b) is chosen from the square R = {0 ≤ a ≤ 1,0 ≤ b ≤ 1}. Suppose that the probability distribution on R is uniform; that is, the probability that a point (a,b) comes from a region
Prove that P(ABc) = P(A) − P(AB), and draw an illustration picture.
How many seven-digit telephone numbers can be arranged if a telephone number does not begin with 0 or 1? Find the probabilities that a randomly selected number contains exactly three ones, exactly
In Example 3.2-3, what is the probability to get the car if you follow the switching-strategy?Example 3.2-3The name comes from a game show host. You are on a game show. There are three doors: behind
Clearly, starting from zero, there are 100,000 numbers which may be written by five digits or less. Show that this also follows from the basic counting principle. How many numbers may be written
You mark five cells from fifty in a lottery ticket. Find the probability that you have guessed all five numbers; exactly three of them.
You select at random k numbers from the sequence 1, ...,n. Show that the probability that you will choose a particular combination of numbers, say, 1,2, ...,k, is 1/ (n k). Suppose that now you
Similar to Example 2.2-7c, prove that the probability of selecting a red ball at any draw is the same as at the first draw. Example 2.2-7cThere are exactly two participants in a contest that
In Bridge, fifty-two cards are dealt to four players, thirteen to each.(a) Write a formula for p1, the probability that a particular player, say, the first, will get a whole suit (all thirteen cards
There are ten pairs of shoes in a closet.(a) If you choose two shoes at random, what is the probability that it will be a pair?(b) You have chosen four shoes at random. Find the probability that
Let P(A1) = P(A2) = 1. Show that P(A1 ∪ A2) = P(A1A2) = 1.
Among 200 items, there are 10 defective. If you choose at random 20, what is the probability, that all will be non-defective?
Each week, on one of the weekdays, Joan receives a flyer advertisement from a particular store. It happened that in four of the last five weeks, the fliers came on Fridays. Given this, to what extent
In a party of ten, each person shakes hands with each. What is the number of all handshakes?
Suppose n cards from a well shuffled deck of 52 are dealt out. If 5 ≤ n ≤ 52, what is the probability that the first five cards are(a) Spades,(b) Red?
Seven apples, three oranges, and five lemons are randomly distributed into three boxes. No box can contain more than five fruits. Find the probability that(a) Each box contains an orange;(b) Exactly
A professor is preparing a final for n students. Each student will be given a theoretical question and a particular problem on calculations. The professor has prepared n theoretical questions and n
Suppose that for each member of a family of five people, all months are equally likely to be that of the birthday. Find the probabilities that(a) All five were born in the same month; (b) All five
Each of one hundred students independently chooses one of five elective courses. Write a formula for the probability that each course will be chosen by twenty students.
An elevator in a ten-floor building leaves the first floor with six passengers. Assuming all possible outcomes to be equally likely, find the probabilities of the following events.(a) Three
A well-shuffled deck of 52 cards is dealt out. Find the probabilities of the following events.(a) The fourth card is a king.(b) Among the first five cards, there are cards from each suit.(c) There
John has five pairs of shoes. If he puts the shoes absolutely randomly into five shoe boxes, what is the probability that(a) Each pair will go to the same box;(b) Each box will have one left and one
Find the probabilities that(a) On the k-th day (k < n), there will be a division that will not have been inspected;(b) On the kth day (k < n), each division will have been inspected;(c) On the (n
There are n ≥ 3 pairs of socks in a drawer, and three pairs are black. Find the probability that two randomly selected socks are black. Does this probability get larger when n is increasing?For
In the city of Pleasant Corner, 20% of houses have a pool, 60%—air conditioning. Suppose that for each citizen of Pleasant Corner, the decisions whether to have a pool and whether to have air
Derive (1.3.4.5) from (1.2.2). n P(Bk.n) = = (x^)^ * = *. n! P k! (n-k)! P^ q (1.2.2)
A system consists of components configured as shown in Fig. 6a; pi is the probability that component i works; the components function independently. Find the probability that a signal will go
Prove that, if A and B are independent, the same is true for Ac and B, Ac and Bc.
(a) Consider two singletons, [ω1] and [ω2]; that is, events containing only one outcome each. Are they independent?(b) Are disjoint events independent?(c) Let P(A1) = P(A2) = 1. Are A1,A2
Two cards are selected at random from a deck of 52. Guess whether the events A1 = {at least one card is a king}, A2 = {at least one card is an ace} are independent. Justify your guess rigorously.
Two dice are rolled. Let X and Y be the numbers appeared; and events A = {X is even}, B = {X+Y is even}, A = {X is divided by 4}, B = {X+Y is divided by 4}. Check for independence the pairs A, B and
(a) Consider the tree in Fig. 4b, Section 2.2. Are the events A={on the second day, the price is larger than 9}, B = {on the second day, the price is smaller than 12} independent?(b) In general, if
Explain why for a fixed n, the events Bk,n in Section 1.2.1 are disjoint, and why the total sum of all probabilities P(Bk,n) must be one. Give a heuristic explanation and show it rigorously using the
(a) Consider two trials. We do not impose any independence condition but assume that all outcomes are equally likely. Our intuition tells us that in this case, trials are independent in the sense
Two dice were rolled. Let A1 = {the first die rolled an even number}, A2 ={the second die rolled an odd number}, A3 ={the sum of the results is odd}. Showthat these events are pairwise independent
Make sure that definition (1.1.4) for n = 3 indeed leads to the definition of mutual independence for three events.
To be hired by a company, an applicant should pass two tests. The first test contains 10 questions, and for each question, the probability of giving a correct answer is p1. The second test contains
9% of people who are given a particular drug experience a side effect. Find the probability that at least two of fifteen people selected at random will have side effects.
Four dice are rolled five times. Write an expression for the probability that exactly three times all four dice will show six.
We toss a coin 100 times and are interested in the probability that there will be at most 50 heads. Explain why this probability is larger than 0.5, while if you toss 101 times, it is exactly 0.5.
In a university, 55% of students are females. Consider the probability that in a class of 100 students there will be at most 55 females. Do you expect that this probability is 0.55? Check your guess
You are rolling a die. One face of the die is painted green, another red, and the rest of the faces are black. Find the probabilities of the following events.(a) In the first three rolls, the die
You roll a regular die six times. Write formulas for the probabilities of the following events.(a) The first three rolls show “six”, the fourth and fifth “five”, and the sixth “four”.(b)
For a family having two children, assuming that all outcomes are equally likely, find the probability that there is at least one boy given that there is at least one girl.
For the tree in Fig. 4b, Section 2.2, find the probability that the price increased on the first day given that it did not exceed 14 on the second day. 10 2/3 1/3 14 FIGURE 4. 8 2/5 3/5 4/5 1/5 (b)
Two dice are rolled. Given that the sum is divided by 3, find the probability of two threes.
In a game of bridge, you did not get spades. Write a formula for the probability that your partner does not have spades too. (Bridge is played with a standard 52-card deck by two pairs of partners;
Find P(A1 |A2) in Example 1.1-6 for both cases. Try to minimize calculations.Now, let A1 = {x1 ≥ 0}, A2 = {x2 ≥ 0}. Figure out whether A1 and A2 are independent for both cases.
Show that if A ⊆ B, then P(A|B) = P(A)/P(B). Find P(A|B) for A ⊇ B, and for disjoint A and B. Explain why the last two answers are obvious.
Consider the binomial tree in Section 1.2.2. Suppose that the probabilities of “moving up and down” on the first day equal 1/2, but on the second day, the price moves up with a probability of p1
In a multiple choice exam, a question has two answers, only one of which is correct. Suppose 80% of students know the answer, and those who do not know choose an answer at random.(a)What is the
In a region, a hiker may come across a rattlesnake. For a mountain area and a randomly chosen day, the probability of this event is 0.02; for valleys, it is 0.01. Joan, when choosing one from these

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