Dice rolling

[sigma]

What are the odds or probability of rolling six die and showing
a) all the same number
b) a different number on each die.

[wildwood]

I'll stick my neck out and say the probability in each case is once per 46656 throws.

Let the barrage begin.

[New Judge]

I�ll begin the barrage!

I make it:

(1) One in 7,776 for any number, or 1 in 46,656 for a specific number.

(2) One chance in 64.8 that all will show a different number.

[sigma]

I understand the odds of 46656 but I would have thought that six die each showing a different number would be higher than 64.8

[R1Geezer]

ok the first die does not matter so we have

1/6 x 1/6 x 1/6 x 1/6 x 1/6 = 1 in 46656 for all the same

and

5/6 x 4/6 x 3/6 x 2/6 x 1/6 = 120/46656 = 1 in 388.8 for all different

[New Judge]

The first roll does not matter.

The second has a five in six (0.833) chance of �success� (i.e. not matching the first die).

The third has a four in six (0.666) chance of success (i.e. not matching either of the previous two).

The fourth has 0.5 chance of success.

The fifth 00.333.

The sixth 0.167.

For all to be different each roll must be �successful� so these odds must be multiplied together and they amount to 120/7776 or 0.015432. Which is 1 in 64.8.

I do not believe the number 46,656 is relevant to this calculation.

The reason why the odds against six different numbers are considerably less than six the same is that the earlier dice have quite a high chance of success (five in six, four in six, etc.) whereas to match all six each die only has a one in six chance of success.

It does not matter whether the experiment consists of one die being rolled six times or six dice being rolled simultaneously.

[R1Geezer]

you're right judge the 46656 in my calc of the all different should read 7776, so we have 120/7776 or 1 in 64.8 as above.

[Inforapenny]

judge, your answers are right but I think the number 46656 is at least relevant to the extent that it represents the total number of values of a six digit number (six throws) counting in base 6 (six sides) - i.e 6^6!

as in...

a) all the same = 6 / 6^6 = 1 in 7776

b) all different (1x2x3x4x5x6) / 6^6 = 720 / 46656 = 1 in 64.8