C1. A collector wants to collect football stickers to fill an album. There are n unique stickers to collect.Each time the collector buys a sticker, it is one of the n stickers chosen independently uniformly atrandom. Unfortunately, it is likely the collector will end up having "swaps", where he has received thesame sticker more than once, so he will likely need to buy more than n stickers in total to fill his album.But how many?(a) Suppose the collector has already got j unique stickers (and some number of swaps). Let X, bethe the number of extra stickers he buys until getting a new unique sticker. Explain why X, isgeometrically distributed, and state the parameter p = p; of the geometric distribution.(b) Hence, show that the expected number of stickers the collector must buy to fill his album isnnk=1(c) The Euro 2020 sticker album required n = 678 unique stickers to complete it, and stickers cost15p each. Using the expression from (b), calculate the expected amount of money needed to fill thealbum. You should do this calculation in R and include the command you used in your answer.(d) By approximating the sum in part (b) by an integral, explain why the expected number ofstickers required is approximately n log n, where log denotes the natural logarithm to base e.

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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A box contains r red balls and b blue balls. One ball is selected at random and its color is observed. The ball is then returned to the box and k additional balls of the same color are also put in the box. A second ball is then selected at random, Its color is observed & it's returned to the box together with k additional balls of the same color. Each time another ball is selected, the process is repeated. If four balls are selected, whats the probability that the first three balls will be red and the fourth be blue?
A product shipment contains 6 computer chips, of which two (2) are defective. If an inspector selects 4 chips at random for an order, without replacement, how many ways are there of picking both defective chips in this order (of 4 chips)? Order does not matter.
A psychology lab doing research in dreams has 4 rooms (1,2,3, and 4), each containing 2 beds (A and B). Three pairs of twins are being studied tonight. Each pair of twins will be assigned to one of the four rooms, and then each twin will be assigned a specific bed in the room. How many ways are there to organize the experiment? (i.e. How many ways are there to assign the 6 subjects to rooms and beds if each pair of twins must share the same room?) 24 8 192 48 12
A professor has 15 books. They wish to distribute 4 books each to their 3 favorite students (so 12 in total). How many ways are there to do this? (All 15 books have different contents. We don’t care about the order of distribution, only who gets which books in the end.)
d. [Permutation] Solve the following problems - Find the number of words that can be formed by using the letters of the word "DEMOCRACY" where first letter will be always C and last letter will be R. e. [Combination] A group of 15 students is to be formed from 44 students of MATI00 Course. Among the students of the course there is 2 renowned batsmen, 3 spinners, 2 fast blowlers and 2 wicketkeepers. How many different ways this can be done so as always to (i) include 1 renowned batsman, 1 fast bowler, 1 spinner and 1 wicketkeeper; and (ii) exclude spinners? 2.
In a gambling game, I roll two dice and win $1 if the sum of my two rolls is largerthan 7. I lose $1 if the sum of my two rolls is smaller than 7. In any other cases, Idon’t lose or win anything. I am going to repeat this game 72 times. c) Now make a box model for the number of times I win, including the values onthe tickets, the number of repeats of each ticket, and how many draws we willmake with replacement from the box. The average of the tickets in the box from (c) is ___________ and the SD is___________.
I have a cup with 5 red M&Ms, 4 yellow, and 3 green ones. I draw six at random (without replacement, order does not matter).How many ways are there to draw six at random which give two of each color? For this problem, it can help to imagine that the M&Ms are numbered 1 - 12, with number 1-5 red, 6-9 yellow, and 10-12 green. After that, proceed like we did with Poker hands.
One package contains 5 blue, 4 red and 3 green marbles. We pay $ 14 to draw two marbles at random and without replacement. If we get two blue marbles a prize of $ 20 is won, with two red marbles a prize of $ 30 is won, with two green marbles a prize of $ 30 is won, with two green marbles a prize of $ 50 is won and with any other result they lose the $ 14 they paid to play. Find the expected value of the win in this game. Write your answer in dollars and cents without the $ sign Example: -5.06
Bowl A contains 2 red chips; bowl B contains two white chips; and bowl C contains 1 red chip and 1 white chip. A bowl is selected at random, and one chip is taken at random from that bowl.
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Four salesmen play "odd man out" to see who pays for lunch. They each flip a coin, and if there is a salesman whose coin doesn't match the others he pays for lunch. To clarify, the "odd man" must get heads while the other three get tails OR he must get tails while the other three get heads. Of course, since it is possible for two men get heads and two men get tails, not all flips will result in finding an "odd man". If this occurs the salesmen would be forced to flip again. What is the probability that there is an "odd man" the first time they flip?
You are given a deck of n cards, which are numbered 1 through n. After the n cards are randomly shuffled, the cards are dealt face up on the table, one card at a time. Card rule: after the first card is placed on the table, each new card must have a higher number than the previous card. If it does, this new card remains on the table. If the new card is lower in value, then this card is removed from the table and the game is immediately over. ***The image is the example*** Questions: Prove that for all positive integers n ≥ 3, if there are n cards in the deck, you score exactly 1 point with probability 12 and exactly 2 points with probability 13.

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