A famous problem of probability, where a person samples a set with replacement until every element of the set (i.e. each coupon) has been obtained at least once. Questions deal with the associated probability distribution, proof techniques, etc.

# Questions tagged [coupon-collector]

308 questions

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### Expected time to roll all 1 through 6 on a die

What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die? The order in which the numbers appear does not matter.
I had this questions explained to me by a professor (not math professor), but it…

eternalmatt

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### On average, how many friends would I need to have to have at least one friend's birthday every day?

I know that because of the birthday problem, even after 365 friends, you're going to have a lot of doubles and that there's also an infinitesimal chance that even with infinite friends that there's one day left out. But I was curious how many…

Rohit

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### Expiring coupon collector's problem

The well-studied coupon collector's problem asks, given $N$ different coupons from which coupons are being drawn with equal probability and with replacement:
How many coupons do you expect to need to draw before having drawn each coupon at least…

mjqxxxx

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### Probability distribution in the coupon collector's problem

I'm trying to solve the well known Coupon Collector's Problem by explicitly finding the probability distribution (so far all the methods I read involve using some sort of trick). However, I'm not having much luck getting anywhere as combinatorics is…

Spine Feast

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### How often was the most frequent coupon chosen?

In the coupon collector's problem, let $T_n$ denote the time of completion for a collection of $n$ coupons. At time $T_n$, each coupon $k$ has been collected $C_k^{n}\geqslant 1$ times. Consider how often the most frequently chosen coupon, was…

Did

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### Birthday-coverage problem

I heard an interesting question recently:
What is the minimum number of people required to make it more likely than not that all 365 possible birthdays are covered?
Monte Carlo simulation suggests 2287 ($\pm 1$, I think). More generally, with $p$…

Isaac

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### Random sum in coupon collection

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 the number of unique coupons be $N$, numbered from…

Tony Hellmuth

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### Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$

I've been trying to prove
$$\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$$
I've tried perturbation and inversion but still nothing. I've even tried expanding the sum to try and find some pattern that could relate this to the harmonic…

Transcendental

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### Infection in a village

Consider the following problem:
Suppose a lonely wanderer infected with a virus came into an isolated village with $M$ villagers and stayed there. Every week each of the infected villagers coughs onto $n$ random other villagers (each of them chosen…

Chain Markov

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### Average number of days to see all possible cards

My father and I go to the restaurant everyday, and each one of us needs to grab a card, which has a number from 1 to 600. I thought about registering every new card we see in a list, and a question arose: "How many days, on average, would we need to…

ordptt

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### A number of die rolls to see every number at least once

We have a fair die that can produce $n$ different numbers. How many times should we roll the die to see every number at least once with probability $p$?
Not a homework, just interesting. Tried to solve myself but with no luck.
I think it could be…

DrTyrsa

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### Four balls with different colors in a box, how many times do I need to pick to see all four colors?

I have one white ball, one yellow ball, one red ball, one black ball. I put the four balls in a nontransparent box. I pick a ball from the box to see its color and put it back to the box.
Assuming picking is random, how many times on average do I…

Gqqnbig

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### The coupon collector's most collected coupon

Suppose a coupon collector is collecting a set of $n$ coupons that he receives one-by-one uniformly at random.
If the collector stops exactly when the collection is complete, we know the expected number of coupons in his collection is $n*H[n]$. What…

Brocellous

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### Expected number of types of coupons collected after $n$ picks

There are $k$ types of coupons. Independently of the types of previously collected coupons, each new coupon collected is of type $i$ with probability $p_i$ s.t. $\sum\limits_ {i=1}^kp_i =1$.
If $n$ coupons are collected, find the expected number of…

pan pan

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### Coupon Collector's problem with unequal probabilities

Given 4 different items, each with different chance of being selected, select an item, replace it and select another.
What is the expected mean number of selections that is required to select at least one of each item.
Through some research, this…

James Webster

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