What Is the Probability of Rolling At Least Two 6's in 13 Dice Rolls?

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SUMMARY

The probability of rolling at least two 6's in 13 dice rolls can be efficiently calculated using the complement method. Instead of directly calculating the probability of getting two or more 6's, it is simpler to find the probabilities of rolling 0 or 1 six and subtracting this from 1. The total number of outcomes for 13 rolls is 6^13, and the calculations for the complement involve determining the probabilities for 0 and 1 six, which are significantly less complex than calculating for multiple occurrences.

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Homework Statement



A die is rolled 13 times. What is the probability of at least two 6's appearing? (Round your answer to four decimal places.)

2. The attempt at a solution
I know that the total number of outcomes is 6^13. I did (13nCr2)x(1x1x6x6x6x6x6x6x6x6x6x6x6)/6^13, but the answer isn't right. What am I doing wrong?
 
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Perhaps you should try explaining the theory behind your solution.
I think working on the complement would be much easier. That is, finding the number of times a 6 would appear 0 times and the number of times a six would appear 1 time, and then take the complement.
 
What is the difference between "at least two 6s appear" and "exactly two sixes appear?" which did you calculate?
VeeEight's suggestion is spot on.
 
statdad said:
What is the difference between "at least two 6s appear" and "exactly two sixes appear?" which did you calculate?
VeeEight's suggestion is spot on.


I see. I calculated the number of times exactly two sixes would appear. Not quite sure how to do the compliment though.
 
You can do separate cases (exactly two sixes, exactly three sixes, etc) and add them all up. It's just that taking the complement is less work
(a good exercise might be to do both to make sure they are the same)
 
The complement of "at least two sixes" is "0 or 1 sixes" and is easier to calculate sinced it involves only two cases rather than 5.
 
Thanks! Got it.
 

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