47 votes
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 ...
Mike Earnest's user avatar
  • 76.2k
46 votes
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 $k$ ...
joriki's user avatar
  • 238k
42 votes

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 ...
PseudoNeo's user avatar
  • 9,749
38 votes

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 ...
Vincent Batens's user avatar
37 votes

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 ...
N. F. Taussig's user avatar
31 votes

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 ...
kimchi lover's user avatar
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29 votes
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 ...
Especially Lime's user avatar
27 votes

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 $1/4$.
Empy2's user avatar
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25 votes

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 ...
joriki's user avatar
  • 238k
24 votes

Do I have a misconception about probability?

The average distance (in steps) from where you started is approximately $\sqrt{\frac{200}{\pi}}\approx 7.978845608$ and as the number of steps $N$ tends to infinity is asymptotic to $\sqrt{\frac{2N}{\...
plm's user avatar
  • 1,515
22 votes

Do I have a misconception about probability?

I suspect it comes from the following plausible-sounding but flawed reasoning: (1) the average final position is the starting point (correct) (2) so the distance from the starting point to the average ...
Don Hatch's user avatar
  • 1,047
22 votes
Accepted

Symmetry in Probability (AMC 12A 2023)

The idea in Solution 1, although not explicitly stated, is that if the frog has not reached or passed $10$, irrespective of how much further Flora needs to go, the probability that the next jump ...
heropup's user avatar
  • 137k
22 votes
Accepted

If $(a,b,c)$ are the sides of a triangle, what is the probability that $ac>b^2$?

Assume that the circle is the unit circle centred at the origin, and the vertices of the triangle are: $A(\cos(-Y),\sin(-Y))$ where $0\le Y\le2\pi$ $B(1,0)$ $C(\cos X,\sin X)$ where $0\le X\le2\pi$ ...
Dan's user avatar
  • 22.8k
21 votes
Accepted

More direct approach to a combinatorial problem

Lay down all the copper and gold coins. The probability that the gold coin is first is $\frac1{n-s}$.
JMP's user avatar
  • 21.8k
21 votes

A weird probability question

The difference in probabilities arises because the two sampling methods are different. To understand why, consider an extreme case where one bowl has a single white ball, and the second bowl has $100$ ...
heropup's user avatar
  • 137k
20 votes

Probability of Sisyphus laboring forever

This is a Galton-Watson process with offspring distribution having the probability generating function $P(s) = \frac1{10} + \frac12 s+\frac25 s^2$. The mean of the offspring distribution is $$\mu := P'...
Math1000's user avatar
  • 37.1k
19 votes

Looking for (overkill) usages of indicator functions

Indicator functions are often very useful in conjunction with Fubini’s theorem. Suppose you want to show: $$\newcommand\dif{\mathop{}\!\mathrm{d}} \int_Y \int_{X_y} f(x, y) \dif x \dif y = \int_X \...
Stef's user avatar
  • 1,951
18 votes
Accepted

Union of two events is at least as likely as the product of the events' probabilities

The short answer is yes, and I believe that the reason is more fundamental than you suggest. Since $P(A\cup B)\geq P(A)$ and $0 \leq P(B) \leq 1$ it follows quite naturally that $P(A) \geq P(A)\times ...
Red Five's user avatar
  • 2,016
17 votes
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 ...
zhoraster's user avatar
  • 25.5k
17 votes

Probability question about cherry picking

The last $3$ picked (in order) must be red, purple, purple. The simplest solution is to see that by symmetry, P(last $3\; RPP$) = P(first $3\;PPR$), and hence $Pr = \frac8{13}\frac7{12}\frac5{11} = \...
true blue anil's user avatar
17 votes
Accepted

How to reasonably estimate the probability of your father being exactly 12222 days older than you?

Oops, you do say that you're from Belgium (I somehow missed that at first), but I'm going to use statistics from the U.S., because that's what I found first. If your brother turns $10\,000$ days old ...
Brian Tung's user avatar
  • 34.2k
16 votes
Accepted

Probability question about cherry picking

The approach in the question considers all $13!$ permutations of different cherries, but only misses the $2$ ways to order the two purple cherries left in the bowl. The probability is then $$\frac{5\...
peterwhy's user avatar
  • 22.3k
16 votes
Accepted

The probability that ${x_1}^{1/k_1}+{x_2}^{1/k_2}+\dotsb+{x_n}^{1/k_n}$ is less than $1$ - combinatorial proof?

I realized the answer soon after writing the question. Since the site explicitly encourages people answering their own question, I figured I would write it up. First, notice that if $x_1,\dots,x_k$ ...
Akiva Weinberger's user avatar
16 votes
Accepted

Intuition is silent: Find the probability that the smallest circle enclosing $n$ random points on a disk lies completely on the disk, as $n\to\infty$.

First, let me state two lemmas that demand tedious computations. Let $B(x, r)$ denote the circle centered at $x$ with radius $r$. Lemma 1: Let $B(x,r)$ be a circle contained in $B(0, 1)$. Suppose we ...
abacaba's user avatar
  • 8,475
15 votes

Sending a message in bit form, calculate the chance that the message is in its original form after transfer

If this is meant to be practical question, where a good approximation is sufficient, then yes your computation is good! However, it is not the exact answer. As leonbloy pointed out, the case where all ...
Willow Wisp's user avatar
  • 1,163
15 votes
Accepted

Turning over blue and pink cups until the gender of the baby is known.

I don't see an issue with your analysis, and when I calculate $P(E_{15})$ according to your formula, I get $$ \frac{\binom 70 \cdot 0!\cdot 8\cdot 14!}{15!} = \frac{1\cdot 1\cdot 8\cdot 14!}{14!\cdot ...
Arthur's user avatar
  • 200k
15 votes

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

Your question is scratching the surface of the area in probability called percolation theory. Indeed, noting that the connectivity of a random graph in OP's model is independent of the scale, we may ...
Sangchul Lee's user avatar
14 votes

Roll a dice infinitely many times, what is the probability of getting a 5 before a 6

Different method:$\displaystyle P(A_n)=\frac{1}{6}\left(\frac{4}{6}\right)^{n-1}$, where $A_n$ is the event that you get a 5 in the n-th roll and that you haven't gotten either 5 or 6 before the n-th ...
bb_823's user avatar
  • 2,144
14 votes
Accepted

Probability that $b^2 - 4ac \geq 0$ where $a,b,c$ are normally distributed (numerical integration)

Suppose we knew the expected number $\Bbb E$ of real roots of a random quadratic polynomial $aX^2 + bX + c$, where all $3$ coefficients have standard normal distribution, $\mathcal{N}(0, 1)$. Then, if ...
Travis Willse's user avatar
13 votes
Accepted

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

From the original question, with emphasis added by me: Find the probability that the numbers on the slips, in the order he picks up, are in ascending order. From your attempt: picking up 4 slips ...
David K's user avatar
  • 98.7k

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