Six fair 6-sided dice are rolled. What is the probability of getting a "Long Straight", or all six faces showing?

Any help would be great. Thank you.

Guest Nov 3, 2018

#1**+1 **

\(\text{A long straight can appear }6! \text{ different ways}\\ \text{There are }6^6 \text{ total possible different dice rolls (dice are distinguishable)} \\ P[\text{long straight}]=\dfrac{6!}{6^6} = \dfrac{720}{45656} = \dfrac{5}{324}\)

Rom
Nov 3, 2018

#2**+1 **

Rom: Thank you for this. How much more complicated would the problem become if we rolled 10 dice instead of 6 and expecting all faces to show?

Guest Nov 3, 2018

#6**+1 **

It does become fairly complicated with the roll of 10 dice!. However, there is this "general formula" that gives the correct answer to any number of dice => 6. The reason for subtracting and adding is because of "overcounting" and "undercounting" as is discussed extensively here:

**https://web2.0calc.com/questions/probability_882**

So, the actual formula looks like this:

1 - (6 nCr 1 *(5/6)^n - (6 nCr 2*(4/6)^n) + (6 nCr 3*(3/6)^n) - (6 nCr 4*(2/6)^n) + (6 nCr 5*(1/6)^n)), where n=10 in this case. When plugged into the above formula, the result is =0.2718121285, or 27.18% probability.

Guest Nov 4, 2018

#8**+1 **

Hi Melody,

This guest post isn’t mine. This is Mr. BB(_{2}). I suppose I could imitate him using a “monkey see, monkey do” method. To make it more natural, I’d need to eat a few pot-laced brownies and chase them with banana daiquiris infused with 1800 imperial minims of over-proof rum. But I usually just stay sober and troll the BB’s typical dumbness. Here, JB trolls this BB by using my proxy: https://web2.0calc.com/questions/dice-help#r2. JB did an excellent job of throwing this BB off the Troll’s bridge.

This post is much better than his usual inept, sloppy fare. He actually replies with a coherent and usable formula giving its reference, instead of a general equation with an incoherent narrative of BS describing how he derived his answer. Mr. BB has referenced this post before, here: https://web2.0calc.com/questions/the-minimum-number#r3 Apparently he has a great affinity for Naus’ generating function leading to a* modified* Sterling number.

Nauseated’s presentation is a wonder, for sure!

GA

PS You're welcome

GingerAle
Nov 6, 2018