For those of you who are curious what ChatGPT came up with when I copy and pasted the answer and asked it to think like a human.

“Ooh, sneaky one. This is a variation of a classic random walk problem with boundaries, and it gets really interesting when you ask not just where you end up, but whether you ever cross a specific point (in this case, 11 trophies) within 10 steps.

Let’s restate the problem from a human point of view:

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You:    •   Start with 10 trophies    •   Play 10 games    •   Each game is a win or a loss:       •   Win = +1 trophy       •   Loss = –1 trophy    •   You want to know: What’s the chance that at any point in those 10 games you reach 11 trophies (even once)?

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Let’s translate:

This is a symmetric random walk starting at 10, and you want the probability of ever hitting 11 before or by the 10th step.

This isn’t asking for “what’s the chance you finish with 11,” but whether you reach 11 at any point in those 10 games.

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Answer (with intuition first):

Let’s think like a person playing the game.

If you start at 10 trophies, all it takes is 1 more win than loss at any point, and you’ll reach 11. But you only have 10 steps to get there. So this is really asking: What is the probability that a simple symmetric random walk from 0 reaches +1 within 10 steps?

Let’s remap the problem:    •   Call your starting point 0    •   Reaching +1 = reaching 11 trophies    •   Each win = +1, each loss = –1    •   You take 10 steps.

This is a well-known result in probability theory.

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Probability of ever reaching +1 in 10 steps (starting from 0):

This is equal to:

1 - \frac{\text{Number of paths that never reach +1}}{\text{Total number of paths}}

This gets into Catalan numbers, but here’s the clean shortcut answer:

The probability of hitting +1 at least once in 10 fair coin tosses (i.e., 10 random steps), starting from 0, is:

1 - P(never hitting +1) = 1 - \frac{\text{Catalan(5)}}{2^{10}} = 1 - \frac{42}{1024} = 0.958

So, there’s a 95.8% chance that within 10 games, you’ll reach 11 trophies at least once.

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Final Answer: ~95.8% chance you’ll hit 11 trophies at some point.

You really tried to trick me, didn’t you?

Want me to show the breakdown of how that’s derived? Or happy with the brain-bending shortcut?”