# A game with numbers [closed]

Two people randomly select two different 1-digit numbers (including 0 of course) and calculate their absolute difference. Then They share what they get, What is the probability of having the same absolute difference for these two?

For example, Player 1 take "0", then take "3" (he cannot choose 0 again), the absolute difference is 3. Players 2 takes "9" then takes "6", the absolute difference becomes 3. and they share their result of the difference. since both has the same result they yell "yay"... :)

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## closed as off-topic by PythonMaster, Engineer Toast, Spencerkatty, Zandar, AJLMar 5 at 12:12

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is off-topic as it appears to be a mathematics problem, as opposed to a mathematical puzzle. For more info, see "Are math-textbook-style problems on topic?" on meta." – PythonMaster, Engineer Toast, Spencerkatty, Zandar, AJL
If this question can be reworded to fit the rules in the help center, please edit the question.

At some point this site will need guidelines to distinguish math puzzles from math problems. – user1717828 Feb 27 at 13:24
@user1717828 puzzle means "a game, toy, or problem designed to test ingenuity or knowledge." this kind of probability question is actually a puzzle! – Oray Feb 27 at 14:00
So suppose the original numbers were 1 and 2. The absolute difference is 1. What does "They share what they get" mean? – Lawrence Feb 27 at 14:15
@Oray, Consider this math problem (or is it a puzzle?): A red person and a blue person each have an urn filled with 80 red balls and 20 blue balls and alternate drawing one ball at a time. If a person draws a ball of their own color, they can remove 4 random balls from their urn. What is the probability the red person will empty their urn first? IMO, this is just an undergraduate probability question; not a puzzle. – user1717828 Feb 27 at 14:26
@user1717828 since this is not just a straight forward probability question and it requires not only checking probabilities but also some understanding, counting and logic thinking (especially if this problem was prepared with 2 or 3 digits), I believe it requires more than just being an undergrad. I intentionally ask this question as simple as possible (1 digit) to get people's attention and their interest... still there is no right answer has come yet. – Oray Feb 27 at 14:30

Each person picks 2 numbers and finds their absolute difference.

There are 9 ways to get an absolute difference of 1 (1-0, 2-1, ..., 9-8).

There are 8 ways to get an absolute difference of 2 (2-0, 3-1, ..., 9-7).

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There are $w(d) = 10-d$ ways to get an absolute difference of $d$, with $1 \leq d \leq 9$.

Total number of digit combinations is ${10 \choose 2} = 45$.

This is also the total number of 'ways': $\sum_{d=1}^{9} (10-d) = 90 - \frac{9 \cdot 10}{2} = 45$.

We require both to pick the same absolute difference. The probability of this is the sum of the probabilities of both picking the same absolute difference, which works out to be $\frac{19}{135}$.

\begin{align} \sum_{d=1}^{9} \left( \frac{w(d)}{45} \cdot \frac{w(d)}{45} \right) &= \sum_{d=1}^{9} \frac{(10 - d)^2}{45^2} \\ &= \frac{900 - 20(45) + \frac{(9)(9+1)(2 \cdot 9 + 1)}{6}}{45^2} \\ &= \frac{19}{135} \end{align}

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The probability either player has a absolute difference of $k$ is $\dfrac{10-k}{\binom{10}{2}}$. For a match, both need to be the same, and so we consider:

$$\sum_\limits{k=1}^9 \Big[\dfrac{10-k}{\binom{10}{2}}\Big]^2$$

The sum of the first $n$ squares is $\dfrac{n(n+1)(2n+1)}{6}$, and so we have:

$$\dfrac{4\cdot9\cdot10\cdot19}{6\cdot9^2\cdot10^2}=\dfrac{4\cdot19}{6\cdot9\cdot10}=\dfrac{19}{135}$$

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The absolute difference $|a_1-a_2|$ for a player has the following distribution of probability:

$$P(|a_1-a_2|=0)=\frac{10}{100}$$

and

$$P(|a_1-a_2|=k)=2\frac{10-k}{100}$$, for $k=1, \cdots ,9$.

Therefore

$P(|a_1-a_2|=|b_1-b_2|)=\frac 1 {100}+\sum_{k=1}^9 \left( 2\frac{10-k}{100} \right)^2=12.4\%$

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sorry but this is not the right answer :( – Oray Feb 27 at 14:00
@Oray, so each person selects numbers whose difference isn't 0? – humn Feb 27 at 14:12
@hmmn they share results of abs difference with each other only, not the numbers. the absolute difference cannot be 0 since the numbers they have to choose are distinct numbers from each other. – Oray Feb 27 at 14:19