# Thread: Probability question with dice

1. ## Probability question with dice

Situation: When you throw 4 dice at once, the chances all the dice will be divisible by 3 is 1/16

Question: Prove the amount of dice you throw is 4.

Can somebody please explain how the amount of dice is 4?????

Here's how I tried understanding: When you throw 4 dice, there are 5 possible situations you can get all the dice divisble by 3:

1. 3,3,3,3
2. 3,3,3,6
3. 3,3,6,6
4. 3,6,6,6
5. 6,6,6,6

Then I counted all the possible situations you can get when you have 4 dice
(1,1,1,1; 1,1,1,2.....) and that adds up to 24.

So I got that when you spin 4 dice the odds are 5/24 that all the dice will be divisble by 3.

2. ## Re: Probability question with dice

For 2, you also have 3, 3, 6, 3; 3, 6, 3, 3; 6, 3, 3, 3

You have to consider each die separately.

Same for the others

The total number of possibilities for 1 die is 6. For 2 dice, it's 6*6, and for 4 dice, you have 6*6*6*6 = 1296 possibilities

Ways to get all divisible by 3 is only when you get 3 and 6.
There are 5 'groups' of them that you got.

3, 3, 3, 3
3, 3, 3, 6; 3, 3, 6, 3; 3, 6, 3, 3; 6, 3, 3, 3
3, 3, 6, 6; 3, 6, 3, 6; 6, 3, 3, 6; 3, 6, 6, 3; 6, 3, 6, 3; 6, 6, 3, 3
3, 6, 6, 6; 6, 3, 6, 6; 6, 3, 6, 6; 6, 6, 6, 3
6, 6, 6, 6

This makes 16 in all.

The probability is then 16/1296

Hm... are you sure the dice is a regular dice? I think if it were a tetrahedron with 3, 6 and two other numbers less than 6, it'd be getting 1/16.

3. ## Re: Probability question with dice

i am sure it is a normal die. Maybe the book made a mistake.

4. ## Re: Probability question with dice

Perhaps the dice are identical and we cannot distinguish between 3363 and 3633. But, I still do not get 1/16 with this assumption!

5. ## Re: Probability question with dice

Maybe they made a mistake with the number 4. Maybe if you throw 3 or 5 dice you will get 1/16??

6. ## Re: Probability question with dice

Originally Posted by jayshizwiz
Maybe they made a mistake with the number 4. Maybe if you throw 3 or 5 dice you will get 1/16??
The number of dice is neither 3 nor 5.
If the dice were not identical, with 3 dice, the answer would be 1/27 and with 5 dice, the answer would be 1/243.
If the dice were identical, with 3 dice, the answer would be 1/14 and with 5 dice, the answer would be 1/42.

Here is a general formula:
** If 'n' non-identical dice are thrown, then the answer to your question is $\inline\displaystyle \frac1{3^n}$

** If 'n' identical dice are thrown, then the answer to your question is $\inline \displaystyle \frac{n+1}{\displaystyle \sum_{k=1}^{n}\binom{6}{k}\binom{n-1}{k-1}}$ with the convention that $\inline \displaystyle \binom{n}{k} = 0$ whenever $\inline \displaystyle k > n$