# Tag Info

Accepted

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

Suppose all $d=6$ sides have equal chances. Let's generalize and find the expected number of rolls needed until side $1$ has appeared $n_1$ times, side $2$ has appeared $n_2$ times, ..., and side $d$ ...
• 327k

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

The original version of this question started life by asking: how many rolls are needed until each side has appeared 3 times Of course, that is a question that does not have an answer as @...
• 7,823

### Continuous Version of Coupon-Collector Problem

This question reminds me of Wilfrid Kendall's dead leaves simulation, which he uses to explain the difference between forward and backward sampling. Given that the problem can be formalised ...
• 106k
Accepted

### Intuition about the coupon collector problem approaching a Gumbel distribution

Below is a bastardized short version of the connection made in the paper by Holst: The connection with the Gumbel distribution is made with the following steps... Viewing the waiting time to collect ...
• 81.4k
Accepted

### Expectation of collecting stickers in groups

Probability problems can be tricky. Whenever possible, reduce them to steps that are justified by basic principles and axioms. Expectation problems get a little easier because you don't have to keep ...
• 327k

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

This is not easy to compute, but it can be done, provided $\binom{m+k}{k}$ is not too large. (This number counts the possible states you need to track while collecting coupons.) Let's begin with a ...
• 327k

### Estimating n in coupon collector's problem

Likelihood function and probability In an answer to a question about the reverse birthday problem a solution for a likelihood function has been given by Cody Maughan. The likelihood function for the ...
• 81.4k
Accepted

### Number of expected turns to get $k$ distinct numbers

Consider a stage in this process where exactly $i$ distinct numbers have already been seen $(0 \le i \lt N).$ "Equiprobable" means that on average, out of every $N$ times this stage is ...
• 327k
Accepted

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

How many unique values can you expect after throwing a dice with k sides? If you only need the expectation value then the answer is relatively simple. We can compute the probability that a specific ...
• 81.4k

### Probability of all points has been sampled after M trials

In these complex situations it's often much easier to simulate the situation and find an approximate probability than to find an exact analytic expression. Here's a simulation in R showing there is ...
• 2,690

### Expected numbers of distinct colors when drawing without replacement

Suppose you have $k$ colors where $k \leq N$. Let $b_i$ denote the number of balls color $i$ so $\sum b_i = N$. Let $B = \{b_1, \ldots, b_k\}$ and let $E_i(B)$ notate the set which consists of the $i$ ...
• 211
Accepted

### Modified coupon collector: killing traitors problem

As pointed out in the comment, this is a special case of the Negative Hypergeometric distribution. So the answer is, on average, $$\frac{m}{m+1}(n-m)$$ innocents will be killed before all $m$ traitors ...
• 421

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

Contrary to the answer by kjetil, this is actually the "classical occupancy problem" (which is related to the coupon collector's problem, but is not quite the same problem). The random variable $|S|$ ...
• 127k
Accepted

• 3,370
Accepted

### Quiz with questions sampled from a pool. How many new questions at each iteration?

Each question has a probability $1-g/n$ of not being picked in each iteration. Hence after the $i$'th iteration (letting $i=0$ denote the first iteration), each question has not been picked with ...
• 11.2k

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

Consider $m=10$ groups of size $n=10$ each. Obtain a sample of size $N$ by withdrawing one element at a time. For the possible counts $0, 1, 2, \ldots, m,$ let $p_N(i)$ be the chance that exactly $i$...
• 327k

### Hypothetical roulette problem

If you want the expected number, then you can use a recursion for an exact calculation (up to precision errors). For example in R ($7$ million loops takes some time): ...
• 40.4k

### How many samples are needed to observe every subject?

This is a variation of the Coupon Collector problem, extensively discussed on this site. However, the algorithm proposed for this variation is based on the Principle of Inclusion-Exclusion, which is ...
• 327k
1 vote
Accepted

### Calculate probability

Let, Y be the number of fake messages sent by trickster for all the original messages to be forwarded. E[Y] be the expected number of fake messages sent. We can write Y = X_1+X_2+\...
1 vote

### A Pairwise Coupon Collector Problem

A Solution for Question 1 \begin{align*} E(X) &= \binom{M}{2}\left(1 - (1-p)^T\right) \\ V(X) &= \binom{M}{2}(1-p)^T\left(1 -\binom{M}{2}(1-p)^T\right) + 6\binom{M}{3}(1-q)^T + 6\binom{M}{4}(...
• 8,317
1 vote
Accepted

### 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?

We could find the probability of not having some items selected at all after the experiment, for a given $m$. For a chosen set of $x$ items, the probability of not having selected them at all after $m$...
• 57.7k
1 vote

### Probability that all cards have been drawn

This is a simple variant of the coupon collector's problem, which uses the classical occupancy distribution. Let $K$ be the number of distinct cards that have been drawn, distributed according to the ...
• 127k
1 vote
Accepted

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

This answer was what I was looking for. With a single cap, I can get the probability that any point chosen randomly from the surface will not be part of the cap as: $Pr=(1 - \frac{m}{n})$ and for $k$...
1 vote

• 20.9k
1 vote

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

Let us reformulate and generalize a bit. Say we have a collection of $n$ objects ($n \ge 1$). We sample with replacement from this collection $k$ times. Each time the probability of selecting object \$...
• 80.6k

Only top scored, non community-wiki answers of a minimum length are eligible