We celebrate five years of the math factor in todays segment!!
Let’s explain a couple of the calculations a little more:
If $100 is bet, then the game will end once one has rolled seven heads; I guess I said that’s eight rolls (thinking that there would then be a tail— but that’s irrelevant: the game would end either way!) So that game is worth 7 * $0.25, or just $1.75.
The second Parondo Paradox game seems like it should be a winning game. If the first coin is tossed 1/3 of the time (winning 40% of the times it is tossed), and the second coin 2/3 of the time (winning just over 55%, say 55.001% of the time it is tossed), then the expected value should be positive:
1/3 x 40/100 x ($1)
+ 1/3 x 60/100 x (-$1)
+ 2/3 x 55.001/100 x ($1)
+ 2/3 x 44.999/100 x (-$1)
Which equals a whopping $1 in 75000 throws. But it’s positive!
But interestingly, the game is slightly weighted towards the first coin, just enough to tip the balance towards a net losing game. Remember, which coin you toss depends on the value of the pot, mod 3, and this in turn depends on the value last time, and what the last toss was, which in turn depends on…
For this version of the game, one expects to use the first coin about 33.5% of the time, and lose, on average, $1 in each 160 throws.
But if we reset the game periodically, it’s much like in the first case and we can still win!