Original Problem and Title-I

I managed to find two perfectly fair coins (don't ask me how, I just did). One of them I gave to my math teacher for Christmas (he was most ecstatic) and the other I kept.

Next Friday, we played a game. I tossed my coin 876 times (it was a great math lesson!) and my teacher, being rather more patient, flipped it 877 times. I would like to know, what would have been the probability of me tossing less heads than my teacher, as a decimal?

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