What are the odds you'll roll a 5 and a 6 in one roll with 5 dice?

Question

I know the odds of rolling one 6 in 5 dice is 1-(5/6)^5 divided by (5/6)^5 which is 3/2. 

But to get the 6 and the 5 would I just do (4/6)^5 instead?

Answer 1

The problem with your idea of just using 4/6 (not getting any 5s or 6s) would not be correct because that excludes the possibility of getting (a 5 and no 6) or (a 6 and no 5).

 

When calculating odds you don't need to calculate probability (although you can use it to get odds). 

 

Odds = (# ways to succeed)/(# ways to fail)

 

For your question   # ways to succeed =  all possibilities - (no 5) - (no 6) + (no 5 and no 6)

 

                                                        =  6⁵ - 5⁵ - 5⁵ + 4⁵ =2550

 

                            # ways to fail  = 6⁵ - 2550 = 5226

 

                            odds of at least one 5 and at least one 6 = 2550/5226

 

If you did calculated probability of getting at least one 5 and at least one 6 say =  a/b and you wanted odds the answer would be a/(b-a).

 

Any questions please comment back.

 

                            

Answer 2

The complement is the event that at least one of 5 or 6 is absent.

 

Let's use the Inclusion-Exclusion principle.

 

P(5 and 6) = 1 - P(No 5) - P(No 6) + P(No 5 and No 6)

 

= 1 - (5/6)⁵ - (5/6)⁵ + (2/3)⁶

 

= 425/1296 ≈ 32.8%

 

 

Answer 3

Short and elegant. Great answer.

 

Only thing is that (2/3) should also have exponent 5:

 

P(5 and 6)=1-(5/6)⁵-(5/6)⁵+(2/3)⁵