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Questions tagged [coupon-collector-problem]

In probability, the coupon collector's problem models "collect all the coupons and win" contests. In its simplest form, suppose n distinct types of coupon are equally likely to occur: what is the expected number of coupons we must draw to collect at least one of each type?

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Coupon collector inside the coupon collector

Consider the coupon collector problem, in which $n$ coupons must be collected and one coupon appears randomly daily. It is well-known that it takes an average of $n H_n$ days to collect all coupons, ...
0 votes
0 answers
10 views

Estimating a subpopulation's size based on distribution

Say I have population $ n_1{} $, where each individual samples is unique and contains multiple traits. A subset of population$ n_1{} $ forms a population that I'm interested in, called $ n_2{} $. Each ...
3 votes
1 answer
34 views

Hypothetical roulette problem [duplicate]

If you have a roulette table that has 7,000,000 places and you have 10 balls (and only one ball can fit onto each number at every roll), how many rolls would it likely take for a ball to land on every ...
0 votes
0 answers
67 views

Total distribution of m geometric distributions [duplicate]

Suppose that you made a webpage and you are collecting the statistics from the visitors. There are m types of visitors (students, company recruiters, etc.). Each visit is equally likely to be any of ...
9 votes
1 answer
431 views

Multinomial distribution conditional on number of distinct items

I want to sample from the integers $\{1, \dots, k\}$ with probabilities $\{ p_i \}_{i=1}^k$, with replacement, until I see $m$ distinct elements (call that $n$ times). You can view the distribution I ...
34 votes
2 answers
7k views

How many stickers do I need to complete my FIFA Panini album?

I'm playing the FIFA Panini Online Sticker Album, which is an Internet adaption of the classic Panini albums that are usually published for the soccer world cup, European championship, and possibly ...
1 vote
1 answer
144 views

How many samples are needed to observe every subject?

I have this problem. I have to analyze 13 chemical substances (n=13) in more than one analysis cycle. In each cycle of analysis, 8 substances are randomly chosen (m=8) to be analyzed. I would like to ...
0 votes
1 answer
203 views

Expected Unique Toys [duplicate]

Suppose there are n types of toys, which you are collecting one by one. Each time you collect a toy, it is equally likely to be any of the n types. What is the expected number of distinct toy types ...
9 votes
1 answer
396 views

Intuition about the coupon collector problem approaching a Gumbel distribution

The coupon collector's problem Let there be $n$ different types of coupons and we try to collect all of the types. We do this by independent random draws of coupons in which each type of coupon has an ...
2 votes
0 answers
25 views

How can I find the expectation value to this problem?

At a wedding reception on an evening the representative of the host is taking it as an occasion to exercise and explain a classical analytic problem. specifically, he insists that he would start ...
1 vote
1 answer
83 views

Calculate probability

Assume that a remailer reorders messages. It has a pool of n-1 messages at all times. When the nth message arrives, one of the n messages is selected at random and forwarded. An attacker fools the ...
1 vote
1 answer
73 views

Probability of all points has been sampled after M trials

If I have a sequential dataset of size 10,000 (e.g. timeseries) and sampling randomly a batch size of 100 random values, each point of the 100 is sampled with the next 5 points after it. so basically ...
2 votes
2 answers
208 views

How many unique values can you expect after throwing a die with k sides?

I ended up asking here. My problem might be familiar with the coupon collector's problem and related to this post Probability of throwing n different numbers in m throws of a die but it does not solve ...
1 vote
0 answers
47 views

How many random draws needed to sample a uniform space?

I am looking for a way to estimate how many random draws took place to produce a produce a known number of unique results. I'm sampling a uniform distribution. For example, if there were a lottery ...
3 votes
2 answers
242 views

Combinatorics - calculating probability of choosing n people from m groups of k people

I am trying to calculate and understand the following question: I have 10 groups of 10 people each and am choosing 10 people at random. What is the probability that all 10 people I have chosen are ...
3 votes
1 answer
86 views

Quiz with questions sampled from a pool. How many new questions at each iteration? [duplicate]

I am taking a quiz which is a sample of $g$ questions from a pool of $n$ total questions. At every iteration $i$ I get a new sample of $g$ questions. I want to know the expected number of new ...
2 votes
2 answers
107 views

Calculate probability that I have fully sampled a set [duplicate]

Let's say I have a set of items out of which I randomly take 5% . Then perform some action on these items and put them back in the set. After how many repetitions of this process can I be 95% sure ...
5 votes
1 answer
2k views

Number of expected turns to get $k$ distinct numbers

We have a random number generator, which generates random numbers from $1$ to $N$. Each number has an equal probability of occurring (equiprobable). Find the expected number of turns to get $k$ ...
0 votes
1 answer
146 views

Inverse coupon collector problem (estimating n), unequal case

I collect coupons of theoretically $n$ different colours, which come up randomly, one at a time. Up to now, I have collected $c$ coupons in total, falling into $j$ different colour classes. From this ...
23 votes
4 answers
5k views

What is a tight lower bound on the coupon collector time?

In the classic Coupon Collector's problem, it is well known that the time $T$ necessary to complete a set of $n$ randomly-picked coupons satisfies $E[T] \sim n \ln n $,$Var(T) \sim n^2$, and $\Pr(T &...
2 votes
1 answer
78 views

A Pairwise Coupon Collector Problem

This is a modified version of the coupon collector problem, where we are interested in making comparisons between the "coupons". There are a number of constraints which have been placed in ...
16 votes
1 answer
2k views

Expected numbers of distinct colors when drawing without replacement

Consider an urn containing $N$ balls of $P$ different colors, with $p_i$ being the proportion of balls of color $i$ among the $N$ balls ($\sum_i p_i = 1$). I draw $n \leq N$ balls from the urn without ...
10 votes
2 answers
5k views

Is there a formula for a general form of the coupon collector problem?

I stumbled across the coupon collectors problem and was trying to work out a formula for a generalization. If there are $N$ distinct objects and you want to collect at least $k$ copies of each of ...
0 votes
0 answers
24 views

A kind of coupon collector's problem [duplicate]

I have been struggling to find and understand the solution for a problem that I have. Here is a link I referenced for this problem. Here's a proposed solution that I was able to find: Let CC(n) = ...
15 votes
3 answers
2k views

Estimating n in coupon collector's problem

In a variation on the coupon collector's problem, you don't know the number of coupons and must determine this based on data. I will refer to this as the fortune cookie problem: Given an unknown ...
20 votes
3 answers
4k views

Estimating population size from the frequency of sampled duplicates and uniques

There is a web service where I can request information about a random item. For every request each item has an equal chance of being returned. I can keep requesting items and record the number of ...
1 vote
1 answer
315 views

How many times I need to draw k items from a list of n items to get a probability P that each item is selected at least once? [duplicate]

I have seen a few related questions but not exactly what I am looking for (in particular this and this) I think. I may be missing something since I am only a beginner at these counting problems. I am ...
6 votes
2 answers
192 views

Probability that all cards have been drawn [duplicate]

Let's say, I have a deck of $m$ cards from which I will draw $n$ cards at random, watch them and put them back in the deck. I want to know; what is the change after $k$ draws, that I have seen all ...
4 votes
1 answer
143 views

Modified coupon collector: killing traitors problem

You have $n$ people, $m$ of whom are traitors. You keep killing them randomly until all the traitors are gone (which you presumably have some way to verify). On average, how many people $X$ will you ...
2 votes
0 answers
201 views

Intuition about a coupon problem were we ask for the distribution of the unique coupons when the number of draws is fixed

Alternative viewpoint of the coupon collectors problem In the coupon collectors problem we draw from a collection of $n$ coupons, with replacement and ask the question how many draws $K$ it takes to ...
43 votes
7 answers
21k views

How often do you have to roll a 6-sided die to obtain every number at least once?

I've just played a game with my kids that basically boils down to: whoever rolls every number at least once on a 6-sided die wins. I won, eventually, and the others finished 1-2 turns later. Now I'm ...
4 votes
1 answer
109 views

Finding a discrete distribution for the minimum number of times needed for all events to occur

Is there a distribution that describes the number of trials before all the events of a random variable with a discrete uniform distribution occurs? Examples: The number of rolls before getting all ...
3 votes
2 answers
94 views

What proportion of the space is taken up by independent discrete uniform variables

If you take $N$ independent uniform random selections from a discrete space with $M$ possibilities (with replacements), then what proportion of the possibilities will have been selected? Formally, ...
3 votes
1 answer
104 views

How many samples do I need to take to expect I have seen (x%, say 99%) of the finite population?

Let's say I have balls numbered 1 to 100 in an urn. Every time I pick one at a random, record it as seen and put it back in the urn. How many such picking do I have to perform if I expect to see at ...
4 votes
1 answer
219 views

Coupon Collection Random Sum [closed]

I have a problem which involves the standard coupon collector's problem to find a probability density from the generating convolution. I start by defining the problem and a few basic statistics. Let ...
1 vote
1 answer
104 views

Need help with expected value calculation in coupon collectors in coupon collectors variant

I am revisiting a question I asked previously with a slight caveat. In my new situation, I am considering the marbles to always be attached to the same neighbors. Hopefully this will be clearer with a ...
2 votes
0 answers
255 views

Extending the Coupon Collector Problem

The Coupon Collector Problem is a well-known one in mathematics and statistics. The problem assumes that all unique coupons are equally likely a priori and gives the expected (mean) number of coupons ...
2 votes
1 answer
639 views

What's the expected number of distinct values within a binomial distribution sample?

Given $X$ has a binomial distribution $B(n,p)$. Now I take $k$ samples from $X$. What's the expected value of how many distinct values I sampled? My rather futile approach was: Given after $k$ draws, ...
1 vote
0 answers
253 views

Understanding the equation for the coupon collector problem

I am collecting the new world cup panini album. The album has 680 stickers (approx) and there are 5 different stickers in each packet. I would like to know the probability I have of missing 50 sticker ...
7 votes
1 answer
1k views

In the coupon collector's problem with group drawings, why does the probability decrease with increasing samples?

My son is collecting Panini stickers from one of their football albums, where there are 472 stickers in total, and you can buy them in packs of 5 (no duplicates within those 5). You can also buy any ...
5 votes
1 answer
4k views

Probability of selecting each item at least once when sampling with replacement [duplicate]

Sampling with replacement $N$ items from $K$, where $N \geq K$, what is the probability that each item in $K$ is selected at least once? For example, from $10,000$ individuals, what is the ...
1 vote
0 answers
74 views

Birthday problem [duplicate]

Recently I read about how many people it takes for the probability that two or more share the same birthday to be sufficiently high. My question: How many people do I have to know that I could be ...
5 votes
1 answer
594 views

Continuous Version of Coupon-Collector Problem

Suppose I get in an elevator every morning and am in it for exactly one minute. A continuous loop of music plays in the elevator and is 10 minutes long. If my daily arrival time is uniform in the ...
1 vote
0 answers
24 views

Probability an binomial experiment terminates

I was recently thinking of how to calculate that an experiment terminates after a given amount of turns. An example would be something similar to opening packs of baseball cards. Suppose there are a ...
15 votes
2 answers
1k views

Expected number of times to roll a die until each side has appeared 3 times

What is the expected number of times you must roll a die until each side has appeared 3 times? This question was asked in primary school in New Zealand and it was solved using simulations. What is ...
5 votes
1 answer
437 views

How many times, in expectation, am I fetching a random sample from a predefined discrete probability distribution of elements?

As part of a simulation that I'm working on, I have a probability distribution over $n$ elements, from which I have to sample a set $S$ of size $m$. That is, each element $e \in S$ must be unique [1]...
1 vote
1 answer
308 views

Probability of Jar containing Balls type question with replacement

A jar has 50 balls 1 to 50 each one having distinct number written on it. Bob, owner of the jar,each day he takes out one ball out of the jar randomly ( with equal probability) and put it back. Q1. ...
5 votes
1 answer
378 views

Probability associated with experiencing all outcomes

Assume last year I commuted to work in a taxi, and suppose if there are $n$ taxis in the fleet I used. If I took one of these taxis every trip at random and with replacement, then what is the number ...
5 votes
1 answer
621 views

Distribution based on rate of distinct values

I am investigating the following problem: I have a large set of values, many of which are repeated. Measuring the number of distinct (or unique) values shows that their number grows much slower than ...
0 votes
0 answers
84 views

coupon collector's problem - labelled or not?

I have come across the famous coupon collector's problem. Linking the problem to the Twelvefold way, my first reaction has been to use the case with ...