Interesting Math

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A Random Player
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Interesting Math

Post by A Random Player » Sat Aug 22, 2015 10:50 am

I'm surprised we don't have a thread like this yet.

Anyway, what inspired me to post this was this puzzle from the xkcd forums:
The game is, the reaper will flip a biased coin. There is a 55% chance of heads, which will double your remaining earthly time, and a 45% chance of tails, which will reduce your remaining earthly time by 70%. The reaper will play this with you as often as you want, but you must state ahead of time how many games you will play. The games will be resolved instantly and exactly according to these rules with no cheating.

So, if you have 10 days left, and you play, you have a 55% chance of ending up with 20 days left, and a 45% chance of ending up with 3 days left. Doing the math, this has an expected value of 12.35 days. Awesome! Each game increases your expected remaining time by over 20%, and the most probable outcome is an increase in time remaining! What could go wrong?

So you declare you'll play the game with the reaper 1000 times. That's not living forever, but it ought to be pretty close! At first it was great, you got three heads in a row and you have months extra! But by the time you're done, your time left on this plane is best measured in Plank units! The reaper offers to reset the game, and play once again with 1000 tries if you feel like you were unlucky. You get even worse results! Indeed, this is a loosing game!
Source, with spoilers
There are no tricks - for all x, the expected value is 1.235x (exactly). Even with multiple possibilities, the expected value for all of them increases, so overall EV increases (the 45% chance of 3 days gives another 45% of .9 days, 55% of 6, similar with the 55% chance of 20 days). However, if you run a simulation, you will indeed get a very small result.
$1 = 100¢ = (10¢)^2 = ($0.10)^2 = $0.01 = 1¢ [1]
Always check your units or you will have no money!

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Re: Interesting Math

Post by AlternateGravity » Sun Aug 23, 2015 12:01 am

The reason for this is that each decrease is a 70% decrease so each decrease causes your life time to be three and a third times shorter while each increase only increases your life time by one half so while decreases are less common then increases each decrease has a significantly larger effect than each increase. 45% is 81.9% of 55% while 2 is only 60% of 3+1/3 so there is a greater percent difference between 2 and 3+1/3 than there is between 45% and 55% so the significance of each decrease is more than enough to make up for the fact that the decreases are rarer than increases. This demonstrates that when comparing risks to potential benefits one should not only look of the probability of risks compared to the probability of benefits but its good to look at the significance of each individual negative consequence if it happens compared to the significance of each individual benefit if it happens.
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Re: Interesting Math

Post by testtubegames » Thu Sep 03, 2015 4:24 pm

I've heard a similar one with coin tosses and money. Start with a dollar. If you flip heads, you triple your money. If you flip tails, you lose all your money. You can play as many times as you want.

If you calculated the expectation values in a similar way to that above - you'd be urged to keep flipping forever. But of course... at some point you'll flip tails and emerge with no money at all.

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