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"I Always See the D@MN Fifth Starter, What Are the Chances?"

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One of the things I absolutely love about living in the city and biking or walking everywhere (other than weekend grocery trips, when a car is very useful), is overhearing very short snippets of people's conversations.  This hit-and-run evesdropping is almost always humorous, given that I have absolutely zero context within which to evaluate statements such as "and then she really kicked him there?" or "if he laughed at you, you really have got to get that checked out."

But last week, I overheard something that I knew exactly what was going on.  And I fought every bone in my body from yelling out some wiseass response while biking by the guy.  So, instead of doing that, I'll post my response here.

Here's what I overheard: "I've been to five games so far this season and have seen the damn fifth starter three times.  What are the chances?"  The Onion hit this topic several weeks back with its perfect story "Phillies Season-Ticket Holder Stuck With Goddamn Joe Blanton Game Again."

So, let's take a crack at it and answer this pedestrian's question.  I think we'll find that his angst isn't really warranted.

Putting aside injuries, day-night doubleheaders, rain outs, etc., the fifth starter will pitch every fifth game, or 20% of the games.  Thus, if you're attending 1 game at random, your chances of seeing the fifth starter are 20%.  (The same is true of your chances of seeing any one particular starter.)  Or, stated conversely, your chances of seeing someone other than the fifth starter are 80%.

Over five randomly chosen games (another assumption here), your chances are simply the product of those percentages -- 3 games at 20% (the fifth starter) and 2 games at 80% (the non-fifth starter).  Multiplying those together, you get a 0.512% chance of seeing the fifth starter 3 times and a non-fifth starter the other 2 times.

But, that's not the answer, because simple multiplication is for one particular ordering -- say, seeing the fifth starter three times in a row and then the non-fifth starter the next two times.  In order to find out the percentage of seeing this mix over the course of any combination of five games, you have to figure out how many combinations there are over five games.

To figure that out, you have to look to the possible permutations over five games.  And, as you can see here (with 5 representing the fifth starter and X representing anyone else), there are 10 different options:  555XX, X555X, XX555, 55X5X, 55XX5, 5X55X, 5XX55, X55X5, X5X55, 5X5X5.

Each one of these has a 0.512% chance of happening.  So, the overall possibility of getting the fifth starter three times and an ace the other two is 10 times 0.512% or 5.12%.**

Thus, 1 out of every 19.5 fans who sees 5 games at random will see the fifth starter three times.  To me, that's not very noteworthy at all, considering 45,000+ fans see a game at CBP every night.

So to the guy I passed on the street last week, here's your answer.  In other words, you're not very special at all.

**Actually, it's a bit higher, as rain-outs, injuries, day-night doubleheaders, etc. increase the pool of "fifth starters" and decrease the pool of aces that people lament not seeing.  So, adjusting for those would put the chances even higher.