Distinction between the probability pre-conditions for a Binomial distribution

Question

For a situation to be modelled using a Binomial distribution, two of the pre-conditions are:

A: The probability of success, denoted $p$, remains the same from trial to trial.

and

B: The $n$ trials are independent.

Now I don't understand how these are distinct.

The way I think about it

• If the trials are independent then $p$ remains constant , if $B$ then $A$.

(I'm not so sure about the next one)

• If $p$ remains constant from trial to trial (the outcome of one trial does not affect the probability of the outcome of any other) then the trials are independent , if $A$ then $B$.

So $A$ and $B$ are equivalent and one is redundant?

Another way to think about it, but I can't come up with an example, is there a scenario where one of $A$ and $B$ could be true and the other false?

Answer 1

A and B are not equivalent and not redundant. Just for an example, think to flip $n$ coins with probability $\mathbb{P}[H]=\frac{1}{k}$; $k =\{2,3,4,...,n-1\}$. (only the first coin is fair...)

If the events are independent but the probability of success is not constant.

If X is the rv "numbers of H" you cannot use a binomial distribution

Answer 2

An example where $B$ is satisfied but $A$ is not: count the number of successful coin tosses when you have two coins, one with $p = 1/2$ and one with $p = 3/4$, where $p$ is the probability of success, and you flip each coin once.

An example where $A$ is satisfied but $B$ is not: count the number of heads in some fixed number of coin tosses when you have two fair coins, but the coins are stuck together so that whenever one lands heads up, the other will be tails up, and vice versa, so although the probability that either coin lands heads is $1/2$, it is impossible to get $2n$ heads whenever you flip the pair of coins $n$ times.