Of trashcans and thimbles.

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.

14 thoughts on “Of trashcans and thimbles.

  1. The last three paragraphs- just amazed how you put the entire puzzle into perspective. Wow.

  2. Loved this post. Statistics never grabbed my interest until I saw Mike LawLers video of him and his kids going through the Monty Hall problem. What I never appreciated about statistics is how reliable a predictor it is, even with small sample sizes. Actually, this is one of the things I love about math: it provides a reasonable organization to what otherwise appears to be random…the trends, the inflection points, and the chances. I’m looking forward to clicking all the links in this post ..and btw I did guess B, thinking my chances or getting the marble aren’t significantly improved by getting just two more tries. And I agree with Radhika’s comment above.

  3. Or as Thomas Aquinas said: Just because an angel is better than a stone, it doesn’t follow that two angels are better than one angel and one stone.

    1. Ooh, good line, Aquinas!

      I’m trying to remember my microeconomic theory from college; as I recall the classical utility function requires a strong assumption about convexity of preferences, which this stone/angel business either directly violates or perfectly exemplifies (need to drink more coffee before I remember which it is).

  4. Of course the better strategy is: Throw tennis balls until you sink one, then switch to marbles. No point in throwing three tennis balls if the first one goes in.

    There’s a different moral there.

    1. True! I originally had a line about needing to select your items in advance, but cut it for space.

      An alternative riddle, assuming you can select items as you go: Which of these three strategies is best?

      A) Throw tennis balls until you make one, then switch to marbles

      B) Throw marbles until you make one, then switch to tennis balls

      C) Alternate between marbles and tennis balls until you make one, then switch exclusively to the one you haven’t made yet

      (My intuition gestures towards (A), but assuming each throw is independent, my intuition is wrong yet again.)

      1. I’ve calculated their probabilities, and they are the same for the three strategies! Also, it is irrelevant in (C) whether you start with a marble or a tennis ball.

        Very nice riddle.

        1. Exactly! The perspective I like on this: imagine you know the future. You have a list of your future trashcan throws (miss, miss, make…) and a list of your future thimble throws (miss, miss, miss…).

          Will you win? It doesn’t matter which list you begin with, or how you move between the lists. It only matters whether the two distances to “make” sum to 10 or less.

    1. Well, as Kev says below, you don’t know how far away this trashcan is! Assuming your tennis ball shooting percentage is a continuous function of trashcan distance, the intermediate value theorem tells us there should be some distance between 1 foot and 1 mile for which your shooting percentage is precisely 50%.

      (Now, I realize the last picture makes it look like it’s about 6 feet, in which case 50% is a pretty sorry shooting percentage, but the last picture also makes it look like the person throwing has no ears or nose, so my cartoons probably can’t be trusted as to-scale diagrams.)

  5. The problem is that we intuitively do not understand what it means that we have to solve BOTH jobs. Or rather, we do not think hard enough. If we did, we’d easily come to the conclusion that there must be a balance between the number of tries for each one. Not trying either of them gives us a no chance of success. The balance, however, is a bit hard to guess. The probability of success for the hard job increases almost linearly with the number of tries. For the easy, it follows an exponential curve. Having to succeed in both with a total of 10 tries means we have to multiply an ascending with a descending curve. The maximum must be somewhere off the middle.

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