Question

The problem of points: independent trials, resulting in a success with probability p and a failure with probability 1-p are performed. what is the probability that n successes occur before m failures?

Answer 1

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The events represented by the terms of your summation are not pairwise disjoint. Consider, for example, the outcome in which the first nn trials are successes; that situation is counted both in the k=0k=0 term and in the k=1k=1 term, since the latter includes the sequence of nn successes followed by 11failure.

We can also use the fact that

∑k≥0(n+kn)xk=1(1−x)n+1∑k≥0(n+kn)xk=1(1−x)n+1

to observe that if p=12p=12, then

∑k≥0(n+kn)pn(1−p)k=12n⋅1(1/2)n+1=2,∑k≥0(n+kn)pn(1−p)k=12n⋅1(1/2)n+1=2,

so your summation must be greater than 11 for sufficiently large mm; this doesn’t explain why it’s wrong, but it does show that it can’t be right.