Ready for a simple probability puzzle with a startling answer?
No, not that one.
Or that one.
Oh, no, I’m not thinking of that other one (I already mentioned it once).
Nah, not those two (which I wrote about ages ago).
Actually, no, not those cool ones you’ve heard about (though I do love those).
No… hey, stop guessing! It’s a different one! Here, let me show you:
So, mull it over. What do you think? (Spoilers follow.)
My intuition insists that (C) must be the correct answer. The tennis ball is easy; the marble is hard; so if I want a snowball’s chance in hell (and, frankly, it sounds easier to toss a snowball into hell) then I need all the marbles as I can get.
But why trust intuition, that impetuous lawyer? The calculation isn’t too onerous. (Said the math teacher.) Here’s the answer in tidy fashion:
Baffling, right?
Or, if you want your bafflement more eloquently put:
Now, is this merely a surprising calculation? Or is there more juice here?
Probability problems often have morals. The birthday problem teaches us that coincidences are commonplace. The classic “false positive” problem teaches us that ordinary evidence should not fully persuade us of extraordinary claims. The Monty Hall problem teaches us that goats aren’t so bad after all.
What do the trashcans and the thimbles teach us? I think it’s this: While chasing your dreams, don’t forget to take out the trash.
See, here’s the basic problem with the 1-and-9 plan. If by some miracle you make one of those 9 thimble throws, there’s still a 50% chance you miss out on the prize, because you botched the one and only trashcan toss.
It’s like picking the winning lottery numbers, but not actually buying the ticket.
It’s like bombing a final-round job interview because you forgot to dress sharp.
It’s like ten thousand spoons, when you should’ve bought 9,990 spoons and 10 knives.
You should trade one of those nine marbles for a tennis ball. That gives a slightly smaller chance of thimble success (8 in a bajllion, instead of 9 in a bajillion) but a much higher chance of trashcan success (75% instead of just 50%). That’s a net improvement: 75% of 8 is bigger than 50% of 9. After that, it’s worth trading another marble for a tennis ball, because 87.5% of 7 is just a bit bigger than 75% of 8. Indeed, as Wyatt Nelson’s nice Desmos explorer demonstrates, this is the best possible combination of throws: 7 marbles and 3 tennis balls.
But the lesson goes deeper. Life has figurative thimbles and metaphorical trashcans.
Take this thimble: Your work goes viral on social media. It’s rare. It’s largely beyond your control. It’s hugely exciting. And it can, on occasion, open the door to some fantastic prizes.
But here’s the trashcan: Are you ready to benefit from the attention? Is your personal website updated? Is your merch store ready? Have you prepped responses to FAQs? When those millions of eyeballs come gazing, are you ready to appeal to the 2-8 eyeballs that really matter?
Every social media post is a marble thrown at the thimble. And those hours spent away from social media, laying the groundwork: those are the tennis balls in the trashcan.
Take your shots at the thimble. But don’t neglect the trashcan.
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