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Let's say I have a regular fair die, but instead of digits 1 through 6, there are three 0s and three 1s.

I can generate a random $n$-digit binary number as follows: if initially I roll a zero (or a series of them), I ignore them; then what is left to do is just roll the die $n$ times. [Thanks to user Alex Meiburg, I have learned that this strategy does generate random numbers, but not uniformly; as he pointed out, generating, e.g., '101' is 3 (!) times more likely than generating '111'. Given this, I would appreciate if someone proposed a strategy lacking this oddity.]

Question: What is the simplest algorithm to generate a random $n$-digit number in base-$b$, $b\neq2$ using this "binary" die?

I have been trying to come up with an answer for a few hours, but I feel that there is none.

user376034
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    The algorithm you describe for binary numbers is random, and has a nonzero chance of producing every n-bit number, but it's not uniform. For instance, you have 3 times the chance of generating "101" over "111". Is this what you want? – Alex Meiburg Nov 19 '16 at 20:34
  • @AlexMeiburg definitely not. I have made an edit pointing this out. – user376034 Nov 19 '16 at 20:39
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    I'm not sure I understand the question. Can you just roll your die $n-1$ times and stick a $1$ at the beginning to maoe an $n $-bit nunber? – Mark S. Nov 19 '16 at 20:50

2 Answers2

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This method is probably best explained by an illustrative example. Suppose you want a random four-digit number in base 6: anything from $1000_6$ to $5555_6$ inclusive (for a total of $1295-216+1=1080$ possibilities).

Since $10$ bits would only give you $1024$ possibilities, we need to use at least $11$ bits, so generate $11$ random bits (by rolling the "die" 11 times), and write them as a binary number (like $00000000110_2=6$). If that number is between $0$ and $1079$ inclusive, add $6^3$ to it to get to get a number between $1000_6$ and $5555_6$. If the binary number was between $1080$ and $2047$ inclusive: too bad, try again by generating 11 new random bits.

In a bad case like this where nearly half of the possibilities are too high to use, the expected number of tries is a little under 2, so this method with retries like this isn't so terrible.

Side note: This algorithm/question is very closely related to the one at Generate random integers from random bit sequence.

Community
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Mark S.
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  • Thank you for this wonderful answer! Side question: does the strategy you described in the question's comments section generate random binary numbers uniformly? – user376034 Nov 25 '16 at 07:46
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    @user376034, If I understand your question correctly, the answer is "yes". Example: If you want a three "digit" binary number, there are $2^{3-1}=4$ possibilities between $100$ and $111$. If you flip a coin (i.e. roll your die) twice then each of "00", "01", "10", and "11" have an equal chance ($\frac14$) of occurring. "sticking a $1$ at the front" as in my comment and "adding $2^2$" as in my answer do the exact same thing: they map these four equally-likely possibilities to the four possibilities you're interested in. – Mark S. Nov 25 '16 at 18:25
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I'm not sure what your difficulty is.

To generate a $n$-digit binary number.

First roll produces $0$ or $1$: this is the first digit.

Second roll produces $0$ or $1$: this is the second digit.

Keep rolling until you have $n$ digits.

This will give you (in binary) the numbers from $0$ to $2^{n}-1$. Each will be equally likely to occur.

To generate a $n$-digit number for another base $b \ne2$, first find the smallest value $k$ such that $2^k >b$.

For the first digit, produce a $k$-digit binary number $n$. If $n \ge b$ then it is not suitable for use in base $b$, so generate another $k$-digit binary number $n$ until $n <b$. This is the first digit.

For the second digit, produce a $k$-digit binary number $n$. If $n \ge b$ then it is not suitable for use in base $b$, so generate another $k$-digit binary number $n$ until $n <b$. This is the second digit.

Continue until you have a list of $n$ digits.

tomi
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  • The first digit of any nonzero binary number is 1. – user376034 Nov 20 '16 at 05:46
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    @user376034 Not necesarily. If I am using 4 bits to encode my binary numbers then the possible numbers are: 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111. These are the 16 different possible 4-digit binary numbers. My algorithm would give each an equal chance of being produced. – tomi Nov 20 '16 at 15:12