Assume that the team plays 8 games. If the team is equally likely to win as to lose each game, what is the probability that they win a string of at least 6 games in a row?

Answer 1

I am assuming that a string of 6 games means 6 consecutive games.

W = win, L = loss

P(W) = P(L) = 1/2

There are 3 ways to win a string of 6 games: WWWWWWLL, LWWWWWWL, LLWWWWWW,

So P(wins exactly a string of 6 games) = 3 × (1/2)⁸

Similarly P(wins exactly a string of 7 games) = 2 × (1/2)⁸

and P(wins exactly a string of 8 games) = (1/2)⁸

P(wins at least a string of 6 games) = P(wins exactly a string of 6 games) + P(wins exactly a string of 7 games) + P(wins exactly a string of 8 games) = 3 × (1/2)⁸ + 2 × (1/2)⁸ + (1/2)⁸ = 6 × (1/2)⁸