Cocktail Napkin Math - What Are The Odds A Fan Catches Two Foul Balls On Consecutive Pitches?
Thinking caps everybody! Thinking caps! It's napkin time.
After helping Dante out with a guesstimate on the odds two Boeing whistleblowers die in a two-month span, I've brought my napkins to map out the craziness from last night's Mariners game.
Wild scene in the left field stands of T-Mobile Park as one single fan out of 14,984 in attendance caught two foul balls on back-to-back pitches. I'm going to go on a limb and say this guy probably never felt so happy having two balls drop right on him unannounced. And that the guys sitting near his section never felt so jealous to see it happen to someone else.
But the real question here is what are the odds of this happening? In baseball. Definitely up my alley to take a stab here. You never know, this might be an interview question you get one day so keep the answer in your pocket.
Anyway, I did a little research to get us started. This Forbes article covered a similar bizarre instance in which Bobby Whit Jr's foul ball was caught by his father (and former big leaguer) Bobby Whit...well... Sr. And Fangraphs estimated catching any foul ball in a game would be 1 in 1000 for an attendance of 30,000 considering about 30 foul balls go in the stands a game.
OK let's do this.
We need to start with the 30 balls hit in the stands per game assumption which we'll consider as fact. It's not consequential one iota that a fan caught a foul ball at some point in the game since it happens 30 times. So safe to say there's a 100% chance some fan catches at least one of the 30 foul balls. That leaves 29 chances for that same person to get another foul ball at some point in the game with an attendance of 14,984 people.
Here's what this all looks like on the napkin.
We're left with a .19% chance any particular fan catches two foul balls at any point in a single game which translates to one in every 526 or so games. Or once every 3-4 years for a given team. I think the real baseball junkies out there could see this as plausible. We all remember a broadcast or two where they ask "hey didn't he already do that?" to a fan in the stands catching a ball.
But we're not done here.
That's all still ho hum stuff. Time to use the odds of this happening at any point in a game to help us find the odds it happens in BACK TO FREAKING BACK PITCHES! I scoured a few estimates for pitches thrown per game in MLB. Seems to be around 300 total for both teams combined. If there's a .19% chance of any fan catching two foul balls at any point at a Seattle Mariners game consisting of 300 total pitches, what are the odds this happens one after the other?
Someone smarter than me feel free to tell me I'm dumb, but I think all we really need to do here is divide the probability of this happening in a game (.0019) by 299 (number of pitches left in the game).
Once more. To the napkin!
1 in 157,480 games.
That's it. That's my estimate. You might even think this is a low figure. I mean, according to baseball-reference there's been 239,986 major league games played as of May 13, 2023. So if the odds of this occuring is one in every 157k games and we've played 239 - wouldn't it be somewhat likely that this has happened before?
I know. Crazy talk, right? Certainly this has never happened before and my estimate must still be way low.
2018:
Everyone happy with my "ballpark" estimate now?
This concludes another segment of cocktail napkin math. Feel free to give me a tip on any future weird randomness you see out there. At least for now you can now showoff in your next interview by reciting the odds of two balls falling on someone.
At least a ballpark.