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We have a story telling dice game, and 3 of the 8 dice have a certain symbol on them (these are 6-sided dice). We roll every dice each time.

I told my son the probability of getting that symbol was 3 in 48 (1 in 16). My husband said it was 1 in 6, as in every throw one of the 3 dice could come up with the symbol.

So basically for the calculation we can actually ignore the 5 dice without the symbol totally, as they have no effect.

I think the 1 in 6 is wrong so I found this page, which says the answer is $\frac{91}{216}$ or 42.13%.

Am I correct in my assumption that we can ignore all the dice which don't have the symbol from the calculation, so it effectively just becomes a rolling a six on 1 of 3 dice question?

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    $\begingroup$ "...the probability of getting that symbol..." When? What is your experiment? Are you throwing 1 dice per time, 3 dice, 8 dice? Define this first. $\endgroup$
    – Jimmy R.
    Commented Apr 4, 2018 at 12:22
  • $\begingroup$ @JimmyR. good point, we throw every dice each time. $\endgroup$
    – WendyG
    Commented Apr 4, 2018 at 12:29
  • $\begingroup$ So, you throw 8 dice (described as above) and you want to know the probability of rolling the symbol at least once? $\endgroup$
    – Jimmy R.
    Commented Apr 4, 2018 at 12:35
  • $\begingroup$ @JimmyR. yes I do $\endgroup$
    – WendyG
    Commented Apr 4, 2018 at 12:42
  • $\begingroup$ Once again my picky comment: "Dice" is the plural of "die". You should say "1 die per time" and "every die" rather than "dice". $\endgroup$
    – user247327
    Commented Apr 4, 2018 at 15:59

2 Answers 2

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Yes, you are correct in your assumption that you can discard all other dice. The correct answer is indeed $42.13\%$. To see this, let $X$ denote the number of times that you roll the symbol. Then, $X$ can take values in $\{0,1,2,3\}$. You want to find the probability $P(X\ge1)$:

\begin{align}P(X\ge 1)=1-P(X=0)=1-\prod_{i=1}^3P(X_i\neq \text{symbol})=1-\left(\frac{5}{6}\right)^3=1-\frac{125}{216}=0.4213\end{align}

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  • $\begingroup$ thanks, i haven't done maths since I was 16 (years ago), so just trying to understand this. Is this symbol ∏ a different rendering of Pi or something else? $\endgroup$
    – WendyG
    Commented Apr 4, 2018 at 12:48
  • $\begingroup$ No, it denotes the product. $X_1$ is the first die, $X_2$ the second and $X_3$ the third. Assume that these are the three dice with the symbol. Then $$\prod_{i=1}^3P(X_i\neq\text{symbol})=P(X_1\neq\text{symbol})\cdot P(X_2\neq\text{symbol})\cdot P(X_3\neq\text{symbol})=\frac56\cdot\frac56\cdot\frac56$$ $\endgroup$
    – Jimmy R.
    Commented Apr 4, 2018 at 12:50
  • $\begingroup$ thanks, i will have to spend time reading this at home after work so in about 8 hours :) $\endgroup$
    – WendyG
    Commented Apr 4, 2018 at 12:58
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You can discard the dice which don't have the symbols on them.

We first calculate the probability of the symbols not appearing on any die, i.e., neither of the die shows a symbol. That will be $\frac{5}{6}$ for one die not showing the symbol. And it will be $\left(\frac{5}{6}\right)^3$ for all of the three dice not showing a symbol.

Since we want at least one symbol, we subtract the probability of no symbols with the total probability, i.e, $1$:

$$ 1-\left(\frac{5}{6}\right)^3 = 0.4213$$

Hence the probability is $42.13\%$

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