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.
+6250 \(\text{A long straight can appear }6! \text{ different ways}\\ \text{There are }6^6 \text{ total possible different dice rolls (dice are distinguishable)} \\ P[\text{long straight}]=\dfrac{6!}{6^6} = \dfrac{720}{45656} = \dfrac{5}{324}\)
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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?
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.
+2539 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 
