How do I (correctly and easily) factor in one trial that has a different probability? For example:

I flip a coin 5 times (50/50). But, on the first flip, I use my special weighted coin that hits HEADS more often, say (80/20).

What are the Odds of each result, # heads: (0,1,2,3,4,5)

Thanks everyone.


You should split into the first flip and then the four trial binomial.

Your probability is then the product of the probability of outcomes from the two separate events, as they are independent.

If we say that the first coin flip is event $A$, and the second is $B$, and with $X$ as the overall outcome as a combination of $A$ and $B$, then $$ \mathbb{P}(X=k) = \mathbb{P}(A=0) \times \mathbb{P}(B=k) + \mathbb{P}(A=1) \times \mathbb{P}(B=k-1) $$

  • $\begingroup$ Thanks Andy, it was the A=0 OR A=1 part that was giving me issues. Appreciate the solution. $\endgroup$ – user3646932 Nov 16 '15 at 15:47

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