TED-Ed lesson:
You’ve been chosen as a champion to represent your wizarding house in a deadly duel against two rival magic schools. Your opponents are fearsome.
From the Newt-niz school, a powerful sorcerer wields a wand that can turn people into fish, but his spell only works 70% of the time.
And from the Leib-ton school, an even more powerful enchantress wields a wand that turns people to statues, and it works 90% of the time.
Lots are drawn, and you’re chosen to cast the first spell in the duel.
The Newt-niz magician will go second, and the Leib-ton enchantress third, after which you’ll repeat casting in that order until only one of you is left.
The rules of magic duels are strict, and anyone who casts out of order immediately forfeits the duel.
Also, to prevent draws, the rules stipulate that if everyone’s still standing at the end of the first round, you’ll all be turned into cats.
Now, you must choose a wand. Your wizarding house presents you with three options: the Bannekar, which binds one target with vines and casts effectively 60% of the time, the Gaussian, which turns one target into a tree and works 80% of the time, and the incredibly rare Noether 9000, which banishes one target to a distant mountaintop and casts perfectly 100% of the time.
Your opponents are masters of strategy, as well as sorcery, and you know they’ll make the choices that maximize their own chances of success.
Which wand should you choose and what strategy should you employ to have the greatest chance of winning the duel?
Pause the video now if you want to figure it out for yourself! Answer in: 3. Answer in: 2. Answer in: 1. You reach for the Noether 9000 first.
After all, it makes sense to enter the duel with the most powerful wand.
But before you pick it up, you consider what would happen.
As the most dangerous wizard, you’d also be the target of the other two magicians, and you’d need to take care of the most dangerous of them first.
But afterward, there’s a 70% chance you’d be struck down by the remaining wizard. That’s trouble.
Maybe it’s better to take the Gaussian. It works 80% of the time, which means you wouldn’t be a target until the enchantress was incapacitated.
But if you succeeded in transforming her, you’d probably be turned into a fish immediately after. If you transformed the sorcerer, the enchantress would almost certainly turn you to stone. It would really be better if you missed.
And that’s when you have an idea: what if you took the Gaussian, then missed on purpose? Then, you would wait for the sorcerer to attack the enchantress, and you’d have an 80% chance of winning against the sorcerer.
It’s a good idea, but there’s a problem. The sorcerer could also pass his turn and the enchantress, knowing that she couldn’t pass without becoming a cat, would cast her spell on one of you.
And since you’re the most dangerous between you and the sorcerer, you’d be the target. And that’s when you see what you really need to do: take the weakest wand, the Bannekar, and miss on purpose.
Now the sorcerer knows that he’ll be targeted by the enchantress and he’ll have to try to turn her into a fish to avoid being turned into stone.
Seventy percent of the time he’d succeed and you’d have a 60% chance of winning the duel at the beginning of the next round.
If he fails, chances are he’ll be turned to stone and you’d still have a 60% chance of winning the duel against the enchantress.
There’s a slim 3% chance you’ll all be turned into cats, but when everything’s accounted for, you have better than even odds of winning with this strategy.
And that’s the best you can do.
Here’s what the probability of winning for the different strategies looks like.
Who would’ve thought that the best way to take your shot would be to throw away your shot?
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