# A Biased Coin Flip Problem

I recently posed the following question on stats.stackexchange.com:

Suppose I have $n$ fair coins, and I mark one of them for identification. Next I flip the $n$ coins without looking. My friend, who is looking on, now informs me that there were at least $k$ landed heads. What is the probability that my marked coin was heads?

$$\frac{\sum _{i=k}^n \binom{n-1}{i-1}}{\sum _{i=k}^n \binom{n}{i}}$$

But suppose we eased the assumption of fair coins, and instead considered the general case where coins could have different (and possibly heterogeneous) degress of bias. Assuming the bias of coins was public knowledge, can we find a general expression to what might now be called "A Biased Coin Flip Problem?"

Intuitively, the probability now depends on how likely or unlikely it is for the other coins to flip heads, and I think the Poisson Binomial may be useful, but I can't seem to work this out.

In case of equal biasing in all coins.

Let, for the biased coin, the probability of landing heads is $p$ and tails is $1-p$. Then if you understood the formula given in question,

The change we need in that formula is only that the numerator needs to be multiplied by the probability of landing head of the marked coin and rest of the formula can be calculated as shown

which is,$$\frac{\text {probab. of heads on marked coin * probab. of getting k-1 heads in rest n-1 coins}}{\text {probability of getting k heads out of n coins}}$$

$$\frac{p(\sum _{i=k}^n \binom{n-1}{i-1}{p^{i-1}}{(1-p)^{n-i}})}{\sum _{i=k}^n \binom{n}{i}{p^{i}}{(1-p)^{n-i}}}$$

(Its derivation can be found here probability of $i$ heads)

In the unbiased case, $p=1-p=\frac 12$ which cancels out in numerator and denominator.

• @ krishna, I don't think that allows for heterogeneous p, or am I missing something? The possibility of heterogenous p's is why I think the Poisson Binomial might be important. Commented Dec 22, 2017 at 13:36
• If you mean different $p$ for different coins, I would fail to do that at the moment.
– sonu
Commented Dec 22, 2017 at 13:39