## April 6, 2011

### On pace for a 162 win season!

Only at the beginning of the season would a four-game winning streak get you an article on S.I.

The probability of a true .500 team playing four games against other true .500 teams and winning them all is 6.25% -- or roughly 2 out of 30. In other words, at this point in the season we could realistically expect two teams to be at 4-0.  Once we consider that in real life teams are not so perfectly matched, thus raising the odds of the more powerful team being successful in all four games, the fact that there are three teams with 4-0 records (Orioles, Reds, and Rangers) isn't a surprise.

The only surprise in all of this is that the Orioles (widely touted to finish last in the A.L. East) swept the Rays in a three game series in Tampa Bay and then went on to beat the Tigers at home in their fourth game of the season. But then again, the probability of a .400 team going 4-0 against a .500* team is just under 3%.  Not very good odds, but something we would expect to see on occasion.

(Shout out to Tango, who raised the "on pace" problem a couple of days ago, and The Book readers who added various lucid and perceptive comments, including an XKCD cartoon. You can never go wrong using an XKCD cartoon to illustrate your point.)

Update: the New Utosky Bolshevik Show takes the Red Sox 0-3 start as its jumping off point for a post titled The Red Sox Aren't Doomed, demonstrating the same thing I did but with graphs and Python script.  Score one for the NUBS.

* Changed from ".600".  Comment #1 below was generated because of this typo; #2 is my detailed response.

-30-

1. I have one minor quibble - A true-talent .400 team against a true-talent .600 team will go 4-0 slightly more than 1% of the time. I assume you took .400^4 and got 2.56% but a .400 team wins 40% of its games against average competition (.500). Raising the opponent's level to .600 means that a .400 team should actually be expected to win roughly 1/3 of the time. Over four games this equates to (1/3)^4 = 1/81 = 1.2%.

2. da5etcetc:

D'oh! A typo on my part in the original (now fixed). I intended to put .500, in large part because it's the easier calculation.

The details, for those following along, is that in cases like this the probabiity of a team winning against a .500 team is its "true talent". That is, a .400 team will beat a .500 team 40% of the time (i.e. .400), a .300 team will beat a .500 team 30% of the time, and a .750 team will beat a .500 team 75% of the time.

The formula for other percentages is as follows, where:
Aw = Team A's winning percentage
Al = Team A's losing percentage (i.e. 1-Aw)
Bw = Team B's winning percentage
Bl = Team B's losing percentage

Probability of A winning against B =
(Aw * Bl)/((Aw * Bl) + (Al * Bw))

Using this formula, a .400 team will beat a .600 team 30.8% of the time (P=0.308). As you say, roughly 1/3.

Being a bit more precise, the probability of the .400 team going 4-0 against a .600 team is 0.308^4 = 0.90%.