if you roll 80 dice, what is the probability of getting the number of 6's equal to 8?

Question

if you roll 80 dice, what is the probability of getting the number of 6's equal to 8?

Answer 1

Hello, thank you for taking the time to post your question!

You can think of this as being a binomial probability question … so for the underlying parameters we have

n = number of trials = 80

k = number of successful outcomes = 8

p = probability of success on a single trial = 1/6

so putting it all together into the binomial formula yields

P(X = 8) = 80choose8 * (1/6)^8 * (5/6)^(80-8)

You can also use something like a binomial calculator to solve too if that makes it easier! Either way the end result here should be

P(X = 8) = 0.0344, meaning that there is about a 3.44% chance of this happening

I hope that helps get you moving in the right direction! Feel free to reach out if you have any additional questions beyond that :)

Answer 2

When we roll a single die , the probability of getting a 6 is 1/6 

and the probability of not getting a 6 is 5/6

We are now rolling 80 such dice and need the probability that we get exactly 8 of the 6's

 

we use the rule that to get the total probability that 'X' and 'y' happens we need to multiply their probabilities.

 

So probability that we get 8 6's is (1/6)⁸

and probability of all other 72 cases being not 6 is (5/6)⁷²

Now we can have the 8 dice to get a 6 chosen in ⁸⁰C₈ ways

 

So the total probability is ⁸⁰C₈ (1/6)⁸ (5/6)⁷²