# Tag Info

Accepted

### Winning strategy for game guessing if next number is prime

More generally, suppose that there are $n$ cards total in the deck, where that $k$ of the cards are considered to be good, while the remaining $n-k$ cards are bad. The goal is the same; the deck is ...
• 76.2k
Accepted

### A disc contains $n$ random points. Each point is connected to its nearest neighbor. What does the average cluster size approach as $n\to\infty$?

This is Stars in the universe - probability of mutual nearest neighbors in disguise. If there are $k$ pairs of mutual nearest neighbours, then there are $n-k$ edges (since $n$ edges are drawn and $k$ ...
• 238k

### Looking for (overkill) usages of indicator functions

Whether it's overkill is open to debate, but I feel that the inclusion-exclusion principle is best seen through the prism of indicator functions. Basically, the classical formula is just what you get ...
• 9,749

### Do I have a misconception about probability?

Since the distance can't be negative, the average distance is strictly greater than zero when at least once you don't land on zero. Let's say you land 2 steps above zero on the first try and 2 steps ...
• 1,849

### Probability that randomly picking $4$ out of $90$ numbers yields an ascending sequence?

What matters here is the order of the slips, not which slips Alex selects. Regardless of which four slips Alex selects, there are $4!$ orders in which Alex could select them. Of these, only one is ...
• 76.7k

### What is the probability that two random subsets of a superset have no intersection?

Construct the sets $A$ and $B$ by following this procedure: for each element of $S$, flip a fair coin to decide if it is a member of $A$, and flip again to decide, independently, if it is to be a ...
• 24.3k
Accepted

### If a deck of 54 cards (including 2 jokers) is evenly split into 3 groups of 18, what is the probability that any one group contains both jokers?

As Shreya says, you are out by a factor of 2. Another way to see this is that $$\frac{\binom22\binom{52}{16}\binom{36}{18}\binom{18}{18}}{\binom{54}{18}\binom{36}{18}\binom{18}{18}}$$ is the ...
• 40.9k

### A question of random points in a square and probability of intersection of their line segments

Let the square be $ABCD$ and the points $P$ and $Q$. Consider the bent lines $APC$ and $BQD$. They intersect once, so by symmetry the chance that $AP$ and $BQ$ cut is $1/4$.
• 50.9k

### A standard 6-sided fair die is rolled until the last 3 rolls are strictly ascending. What is probability that the first such roll is a 1,2,3, or 4?

This can be modelled using a Markov chain. We need $15$ transient states: One initial state in which we have nothing (either because we just started or because the last roll didn’t leave enough room ...
• 238k

• 2,016
Accepted

### On average, how many times must a circular pizza be randomly cut, to get a piece with no curved edge?

Write the expectation as $$\mathbb{E}[X] = \sum_{n=0}^{\infty} \mathrm{P}(X> n).$$ The latter probability is that for $n$ cuts, there is no "central" piece. Note that, thanks to ...
• 25.5k