## Tuesday, June 28, 2011

### Once in a Lifetime

What is the probability that your favorite team will win the World Series during your remaining lifetime? Obviously, the best estimate of this probability depends on the expected quality of your team each year, as well as a specific estimate of your mortality, which is dependent upon any number of factors. This post generalizes this question to estimate the probability of an average team winning the Series during an average person's lifetime.

Before delving into that, I must note that the subject matter is simultaneously frivolous (from a rigid sabermetric standpoint) and a tad morbid. After all, no one really wants to think about their own mortality, and thinking about the chances that your team will break your heart until your heart breaks isn't exactly fun.

I will assume here that the probability of winning the World Series each year takes one of three values, which are independent from year-to-year: 1/30 (all teams have an equal chance), 1/60 (50% as likely to win as average), or 1/15 (twice as likely to win). The last value is also essentially equal to the average chance of winning a pennant (with 14 and 16 team leagues it splits the difference).

What's trickier is figuring the probability of survival at each age. I have used a life table from the Social Security Administration as the basis for these estimates. The table is a projected life table for Americans born in 1980. As such, it doesn't directly apply to people in other age groups, but for the sake of simplicity I have assumed that it does. The effect of this will be to overstate the life expectancy and championship probabilities for people born prior to 1980, as the SSA models assume that the force of mortality will decrease.

Even if the assumptions about mortality prove to be accurate, the probability of your team winning will probably be lower than assumed here thanks to expansion, which may not be imminent but will almost certainly occur at some point.

You can skip ahead a couple of paragraphs if that explanation of the life functions satisfies your curiosity, because this has absolutely nothing to do with baseball. The charts show data for age at five year intervals, starting at age five and running through age 100. They are based only on the male life expectancy chart; females have higher life expectancy, but most fans are male and presenting two separate charts or combining the two would add a lot of trouble to a silly exercise.

There are three pieces of data presented in the chart for each age:

1. e(x)--curtate life expectancy

The curtate life expectancy only considers full-year survival. For example, the life table tells us that at age five there are 98,357 survivors. At age six, there are 98,324. The 33 deaths between age five and six do not contribute to the curtate life expectancy for age five. The complete life expectancy does include partial-years and is included in the life table, but requires the use of assumptions about mortality at fractional ages to estimate. The curtate expectancy is easily calculated from the life table:

e(5) = l(6)/l(5) + l(7)/l(5) + l(8)/l(5) + ... + l(117)/l(5)

= 98324/98357 + 98294/98357 + 98264/98357 + ... + 1/98357 = 73.38

2. Exp Wins--expected championships won by the team during one's lifetime

The probability of a team winning in any given year is assumed to be one of the constants explained above (1/15, 1/30, 1/60). The probability of a person age x surviving to age x + 1 is l(x + 1)/l(x). The product of these two is the probability that a person age x survives to see their team win a title in the year of age x + 1.

So the expected wins for a life age 5 is figured thusly (assuming a 1/30 probability of team winning):

1/30*l(6)/l(5) + 1/30*l(7)/l(5) + 1/30*l(8)/l(5) + ... + 1/30*l(117)/l(5)

= 1/30*e(5)

The assumption here is that one needs to survive the full year in order to see one's team win during that year. To put it in baseball terms, we could say that these calculations assume that each person is born on October X and the World Series always concludes on October X.  I'm also assuming that survival means one is able to enjoy a victory by their team, but sadly this is not always the case.

The use of curtate functions makes computations easier but it also undersells the life expectancy and the championship expectations and probabilities a little bit.

3. >=1 win--the probability that the team will win at least one championship during one's life

For any given age x, the probability that one's team wins and that they survive to see it is 1/30*l(x + 1)/l(x). The complement of this result is the probability that the team does not win the title in this year plus the probability that the team does win the title but life (x) does not survive to see it.

Multiplying all of the complements for a lifetime results in the probability that the team never wins while the subject survives. The complement of this is therefore the probability that the team does win during the lifetime of (x).

Here is the chart based on an average (1/30) chance of winning the title each year:

This paints a pretty rosy picture for most people. Keeping in mind that the life functions are based on expected mortality for those born in 1980, from fifty and younger the expected number of titles is still one or greater, and from 60 and younger the probability of seeing a title is still greater than 50%. If you are indoctrinating your ten-year old son into fandom of your favorite team, you have about a 90% chance of not making his fan life unrewarding, so you can feel good about that.

Of course, the odds aren't as favorable if your team is only half as likely to win as an average franchise:

Still, even if your team can only expect to win once every sixty years, there's no reason to despair. If you are only concerned about winning the pennant, and your team can manage an average probability of doing so, you can feel pretty good regardless of your age:

1. Ok, since 1980, all these teams have won it all--

Phillies (2)
Dodgers (2)
Cardinals (2)
Orioles
Tigers
Royals
Mets
Twins (2)
A's
Reds
Blue Jays (2)
Braves
Yankees (5)
Marlins (2)
Diamondbacks
Angels
Red Sox (2)
White Sox
Giants

...that's 19 teams (covering 31 years).

Do the other 11 teams still have a 50/50 chance of winning it all during the next 28 years? Or does the math alter, as the time limit squeezes shut?

The other 11...

Astros
Brewers
Cubs
Indians
Mariners
Nationals