Rework problem 27 from section 2.4 of your text. Assume that the team plays 7 games.

Question

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 5 games in a row?

Answer 1

P(5 straight after 7 games) with P(w)=1/2 each game

= 6(1/2)^7 = 6(1/128) =3/64= .0234375 .046875= 4.6875%

Answer 2

The question says to win "at least 5" straight games, so we need to include the possibilities of winning 6 straight games and 7 straight games.

There are 3 ways to win 5 straight games:

WWWWWLL

LWWWWWL

LLWWWWW

So P(5 straight games) = (0.5)⁵(0.5)² X 3

There are 2 ways to win 6 straight games:

WWWWWWL

LWWWWWW

So P(6 straight games) = (0.5)⁶(0.5)¹ X 2

There is only one way to win 7 straight games:

WWWWWWW

So P(7 straight games) = (0.5)⁷ X 1

P(at least 5 straight games) = (0.5)⁵(0.5)² X 3 + (0.5)⁶(0.5)¹ X 2 + (0.5)⁷ X 1 = 0.046875