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Aug
27
revised Expected number of tosses for a number to repeat $N$ times given an $n$-sided die.
added 128 characters in body
Aug
27
revised Expected number of tosses for a number to repeat $N$ times given an $n$-sided die.
add note about "reciprocal of the mean" vs. "mean of the reciprocal"
Aug
27
comment Expected number of tosses for a number to repeat $N$ times given an $n$-sided die.
@user245312 - Correction: In my last comment I mistakenly referred to 0.513 as the "binomial probability", when I meant to say it is the expected value $E(N_4)=36⋅p_4$. I was trying to point out that this is consistent with the book, with my answer, and also with your own simulation here. The book appears to treat $\frac{1}{E(N_4)}$ as an approximation of $E(\frac{1}{N_4})$ (the "expected number of trials per occurrence of a $4$-hit box"). The experiment in your most recent comment simulates something very similar, but not quite the same.
Aug
26
revised What is wrong with this logic based on a geometric distribution?
correct the quote
Aug
26
comment Expected number of tosses for a number to repeat $N$ times given an $n$-sided die.
@user245312 - In your last sentence, I think you meant to write that the "expected value of the number of boxes that are hit exactly k times in one trial" is indeed approximately $0.54$ (in agreement with your latest simulation, which verifies the binomial probability of about $0.513$), so a box with exactly $4$ hits occurs approximately $\frac{1}{0.54} \approx 1.85$ times per trial. (More accurately, that's $\frac{1}{0.513} \approx 1.95$ times per trial.)
Aug
26
revised Expected number of tosses for a number to repeat $N$ times given an $n$-sided die.
add [expectation] tag
Aug
26
revised Expected number of tosses for a number to repeat $N$ times given an $n$-sided die.
show the range of $k$
Aug
26
comment Expected number of tosses for a number to repeat $N$ times given an $n$-sided die.
@user245312 - I've added some detail about the expected value calculation. The book appears to focus on the expected value of the number of boxes that are hit exactly $k$ times in one trial, not on the maximum number of hits in one trial. See especially Tabelle 7 in the same section of the book -- it tabulates Poisson approximations to what I have called $E(N_k)$ and $p_k$ for $k = 0,1,2,3,4,5$. (Note that these sum to $36$ and $1$, respectively, over all $k\in\{0,1,...,36\}$.)
Aug
26
comment Expected number of tosses for a number to repeat $N$ times given an $n$-sided die.
@ClementC. - No, there is no need to "restore independence" -- the Poisson approximations are just easier to compute than the exact binomial probabilities.
Aug
26
revised Expected number of tosses for a number to repeat $N$ times given an $n$-sided die.
detail the expectation calculation
Aug
25
revised Expected number of tosses for a number to repeat $N$ times given an $n$-sided die.
add link for proof-method
Aug
25
answered Expected number of tosses for a number to repeat $N$ times given an $n$-sided die.
Aug
23
answered What kind of math is this? Picture
Aug
23
revised How many planar arrangements of $n$ circles?
improve picture for two circles; add descriptions
Aug
23
comment How many planar arrangements of $n$ circles?
Thanks! I've added your second discovery to the question, bringing the number of known cases to $49$.
Aug
23
revised How many planar arrangements of $n$ circles?
add a second new case found by @fuzzy
Aug
22
revised How many planar arrangements of $n$ circles?
update the middle row of the three-circles table with a ninth case found by @fuzzy
Aug
22
comment How many planar arrangements of $n$ circles?
Good find! That's a case I did miss. I'll update the question to include it, making the total now $48$. Mostly just for display purposes, I chose to classify according to the three types of intersection point (rather than how many circles are intersecting there), but there are many alternative schemes.
Aug
19
comment What's the order of growth of the 'double-and-rearrange' numbers?
Ah, I see what you mean.
Aug
19
comment What's the order of growth of the 'double-and-rearrange' numbers?
Are there not exactly $5^n$ $n$-decimal-digit positive integers whose digits are all odd?