Intereting Posts

Syntactically speaking, when can you introduce a universal such that no logical inconsistencies are introduced?
Solving an integral (with substitution?)
Fourier Analysis textbook recommendation
How to reformulate a multiplicative formula (with two primes, perhaps like totient-function)?
taylor series of ln(1+x)?
Integrating $\int\frac{5x^4+4x^5}{(x^5+x+1)^2}dx $
The distinction between infinitely differentiable function and real analytic function
Let $f$ be a continuous function satisfying $\lim \limits_{n \to \infty}f(x+n) = \infty$ for all $x$. Does $f$ satisfy $f(x) \to \infty$?
Evaluation of $\int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx$
$n$ points are picked uniformly and independently on the unit circle. What is the probability the convex hull does not contain the origin?
Counterexample for the stability of orthogonal projections
How prove this inequality $(a^2+bc^4)(b^2+ca^4)(c^2+ab^4) \leq 64$
Each element of a ring is either a unit or a nilpotent element iff the ring has a unique prime ideal
Maximum principle – bounds on solution to heat equation with complicated b.c.'s
Isomorphic subgroups, finite index, infinite index

I was studying the birthday paradox and got curious about a related, but slightly different problem. Lets say I have a set S, that has n unique elements. If I randomly pick k elements from the set (assuming equal probability of picking any element), the probability that I’ll have picked all the elements in the set, when k=n is n!/n^n.

Now, what would be the value of k for probability of picking all elements in the set to be > 0.9 (or some constant).

A simpler variation would be – how many times should I flip a coin to have a probability > 0.9 to have thrown heads and tails at least once.

- Is there some way to simplify $\sum_{i=1}^n \sum_{j\neq i}(\frac{j-1}{2})(\frac{i-1}{2}) $ To obtain a closed form.
- Number of ways of distributing n identical objects among r groups
- Sum of derangements and binomial coefficients
- Why is $S_5$ generated by any combination of a transposition and a 5-cycle?
- Good Book On Combinatorics
- any pattern here ? (revised 2)

- Applications of the number of spanning trees in graphs
- How can I prove this combinatorial identity $\sum_{j=0}^n j\binom{2n}{n+j}\binom{m+j-1}{2m-1}=m\cdot4^{n-m}\cdot\binom{n}{m}$?
- Expectation of Double Dice Throw
- Race Problem counting
- chromatic number of a graph versus its complement
- Upper bound for the strict partition on K summands
- Probability that the equation so formed will have real roots
- Binary matrices and probability
- I roll 6-sided dice until the sum exceeds 50. What is the expected value of the final roll?
- What is the probability of a coin landing tails 7 times in a row in a series of 150 coin flips?

Your first question is about a famous problem called the

Coupon Collector’s Problem.

Look in the Wikipedia write-up, under *tail estimates*. That will give you enough information to estimate, with reasonable accuracy, the $k$ that gives you a $90$ percent chance of having seen a complete collection of coupons.

Hint: If you throw a coin $k$ times and it’s not true that both heads and tails have come up, what must your throws be?

If you’re not sure, try some small value of $k$, say $k = 3$. List all possible throws, and cross out those that have both heads and tails. What remains?

The general probability of having picked all the elements after a sample size $k$ with replacement is

$$\frac{S_2 (k,n) \; n!}{ n^k} $$

where $S_2 (k,n)$ is a Stirling number of the second kind. You want this to be more than 0.9.

For $n=2$, you have $S_2 (k,2) = 2^{k-1}-1$ for $k \gt 2$ so you want

$$\frac{(2^{k-1}-1) \times 2}{ 2^k} \gt 0.9 $$

which is true when $2^{k-1} \gt 10$, so with integers you get $k \ge 5$.

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