Question

Two teams, say the Celtics and the Cavs, are playing a seven game series. The Cavs are a better team and have a 60% chance of winning each game. What is the probability that the Celtics win at least one game? Remember that the Celtics must win one of the

Answer 1

87.04%.

Explanation:

We will focus on how many games it takes for the Celtics to win their first game. Any run of games that ends with a Celtics win counts as a success, as long as it's between 1 and 4 games.

Let `W` be the number of games it takes until the Celtics win their first game. Then `W` is `"Geo"(p = 0.4).`

There is a chance that game 1 is won by the Celtics. The chance of this happening is

`"P"(W = 1) = 0.4`

The chance of the Celtics' first win being game 2 is

`"P"(W = 2) = (0.6)(0.4)`
`color(white)("P"(W = 1)) = 0.24`

The chance of the Celtics' first win being game 3 is

`"P"(W = 3) = (0.6)^2(0.4)`
`color(white)("P"(W = 1)) = 0.144`

Finally, the chance of the Celtics' first win being game 4 is

`"P"(W = 4) = (0.6)^3(0.4)`
`color(white)("P"(W = 1)) = 0.0864`

We stop there, because if the Celtics don't win any of the first 4 games, then the Cavs have won the seven-game series.

Adding all these up gives

`"P"(1 <= W <= 4) = 0.4 + 0.24 + 0.144 + 0.0864`
`color(white)("P"(1 < W < 4)) = 0.8704`