How does this basic probability question solution make sense? I am just having some trouble understanding why it is valid to answer the following question in a certain way.
Suppose we have iid Bernoulli trials, with $p=0.7$,
We do trials until we have 4 success or 4 fail, if we get 4 success first, we win, else we lose
what is the probability that we win.
Now what I dont understand is my, one a work page I had seen, the professor had wrote that we can solve it by noting that it is binomial with 7 trials, and we win if we have 4 or more success. But I dont understand why this is valid because how can we have more then 4 successes, ie , that game is over at that point.
I would think it would be 
=
$$P[S=4, F=0]+P[S=4, F=1] + P[S=4, F=2]+ P[S=4, F=3]$$
But mostly I am wondering about that solution and how it is valid to say they are equivalent even though it does not make sense to have more than 4 successes or failures
Thanks

Your professor not taking 7 trials. He is taking all cases with 0, 1, 2, 3 failures in between successful trials.
