Out of 36 random drawings what are the odds of numbers 1-12 each coming up 3 times?

Question

If you had 12 possible outcomes what are the odds that each outcome would come up exactly 3 times out of 36 drawings?

So I'm going to make a random drawing 36 times and the outcome can be numbers 1-12. What are the odds, that each number would come out exactly 3 times?

 

(note each number is not removed after being drawn, it is put back in after each time)

Answer 1

If the numbers are 1-12 and you are doing the experiment 36 times we can look at the number of possible ways the drawings can occur.

 

N = 12³⁶ since each drawing can happen 12 ways.

 

Now how many satisfy the condition of exactly 3 of each, e.g one such way is. 111222333...(12)(12)(12)  

 

Since we still have 36 objects but this time with duplications how many ways can those objects be rearranged:

 

N_A = 36! / (3!)¹²

 

P(A) = N_A/N is the probability of getting exactly 3 of each.

 

The odds would be (P(A)) / (1-P(A))  (FYI that is the same as (N_A) / (N - N_A) )

 

 

Any questions? Comment back.