The expected value is not necessarily the value you should expect, Simon has proven to me today. It started from a morbid story about a dangerous attraction in Kyoto.
We were walking home from this season’s last outdoor market and Neva told me Jaiden Animations has visited Kyoto and there was “this thing there”. Wonderful, I thought, Neva already knows a place in Kyoto we might want to see one day (I have always wanted to visit Japan)! “It’s a thing you jump from and there’s like a 20% chance that you die!” Neva continued. Ok, we probably won’t be going to see that thing after all. “And Jaiden said someone did it twice!” Neva added. *
Wait, I said. That person had much greater chance to die than if he only jumped once! But what is the way to calculate that?
When I got home, Simon helped me understand how to find out the probability of someone dying if jumping twice. It was one minus the probability of the surviving the first jump times the probability of them surviving the second jump, so 1 – 0.8 x 0.8 = 0.36 or 36%
By how much does your chance to die grow with every jump? Simon started writing everything out in a notebook. “Oh look, those are all the powers of 2!”, he exclaimed. “So if you add all of these up, you get the probability of death if you jump 10 times?” I asked. “Yes, but that’s not what I’m interested in”, Simon said.
It turned out, he was trying to figure out the expected value, i.e. how many jumps on average does one need to do in order to die (I know it sounds awfully morbid, especially for someone who hates Halloween, but hey, it got pretty abstract by this point). I suck at combinatorics and even forgot what expected value was, so Simon explained it to me on the whiteboard.
He started with two coins and 50/50 chance of getting a 1 or a 10, i.e. heads or tails (normal average). he compared that to a weighted coin with 20/80 chance to get a 1 or a 10 (weighted average).
He then returned to our morbid problem, wondering whether there is some smart way to calculate the expected value in a quick fashion. Or is it an infinite sum? If you know a formula to calculate this, please leave a comment!
Simon also shared another example of a coin flipping game, where the expected value is infinity! — The expected value is not necessarily the value you should expect!
In any case, if you’re in Kyoto, don’t be a math prop!
*In the video, Jaiden actually says the probability to survive the “big plunge” is 85%, not 80%. Still not worth it!