Question

Lets's say I flip a coin which a probability p of turning up heads. The game is structured in the way that I win the game if heads appears x times before tails has appeared y times. How can I represent this probability in the form of a summation? In terms of x,y and p.

Answer 1

The probability that we win the game that is head appears x times before the tails has appeared y times is computed here as:

Let there be a total of n tosses in which we obtain x heads. Then this means that there will be (n - x) tails. We want to find the probability that for x heads obtained the number of tails is less than y.

Then, the probability here is computed as:

P(X = x, Y = y) = Probability that we obtain x heads in less than (x + y) total tosses. This therefore is computed here as:

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Note that in each of the expressions on the right hand side of the equation, we want the number of heads to be x and also the last toss should be a head.

This can be now written in the required summation form as:

This is the required expression for the probability function here.