# Is binomial distribution approriate

1. Mar 27, 2014

### aaaa202

Suppose I have 6 die and toss them. The probability to have n 6's is binomially distributed with parameter 1/6.
Now suppose instead tossing the 6 die and having 1/6 probability for a 6 each dice's probability to show 6 grows continously in the time interval t=0 to t from 0 to 1/6. Can I then say that as before, the probability to have n 6's is binomially distributed with parameter 1/6?

2. Mar 27, 2014

### Ray Vickson

I think I know what you are trying to say, but let me re-word it first. You have 6 dice (not 'die'--a 'die' = one single cube). At any toss they each have the same probability p of showing a '6', but the probability p increases from 0 to 1/6 as we make more tosses. If $p_k$ is the probability of a die showing '6' on toss k, then the number $N_k$ of 6's on toss k is binomial with parameters $(6,p_k)$. If we finally get a probability of 1/6 on the nth toss (that is, $p_n = 1/6$), the total number of 6's altogether is $X = \sum_{k=1}^{n} N_k$. This will NOT have a binomial distribution, because although the summands are independent, they are not identically distributed. In fact, you can easily work out the mean and variance of X and find that they are not related to each other as a binomial would give.