Thursday, July 28, 2011

I need to freshen up on my probability!!!

Back on Sunday, I provided my answers to some critical thinking problems. By Monday, I had doubts about my answer to #3.

For reference, here is the text of the problem:
Question 3: While you were at TAM9, you decided to suspend skepticism and gamble - specifically, by playing roulette. But since you want to have some sort of strategy, you decide to flip a coin before each bet to decide whether to place a bet on red or on black (which should have a 50/50 chance of happening). Sadly, you lose sixty seven times in a row - that is, the ball always lands on the opposite color that you pick. If you turned your skepticism back on, it would be most rational to think:

A. You just have shitty luck
B. It's terrible strategy to flip a coin to pick what color to bet on in roulette
C. You should keep up this strategy because you've really likely to win the next bet
D. The roulette table is obviously broken, but you can't assume that's intentional
E. The casino or the staff are dirty crooks who have rigged the game against you somehow
F. You can't reasonably decide which of the listed options are more likely

I was thinking that if you threw in the coin flip, you were reducing the probability to a 1 in 4 chance of getting the desired result. The thought was there is a 1/2 probability on getting a certain coin flip and there is a 1/2 probability on getting a certain color on the wheel. Combining these probabilities (1/2 * 1/2) gets you 1/4. I failed to calculate what the probability of missing 67 times in a row was (it's actually 1 in 235 million, which seems big, but is not necessarily absurd), but I did figure that you should stand to win about 16 times. I figured this wasn't that bad, because probability is no guarantee of an outcome.

However, by Monday (or maybe it was Tuesday...whatever), I was realizing that I was wrong about that 1 in 4 chance. I did this by breaking down the combinations in a truth table fashion (with heads leading me to choose red as a color).

Coin Color* Result
Heads  Red Win!
Heads Black Lose :( 
Tails Red Lose :(
Tails Black Win!
* Represents the color hit, not the guessed color, which is represented by the coin flip (Heads = Red, Tails = Black)

I noticed that two out of the four combinations, I win!!! I used a similar thought process in considering using a 5-sided die to make my guess, where 3 out of 5 times I would be choosing red and 2 out of 5 times I would choose black. I ended up winning half the time.

I now realize where I made my mistake. I was originally thinking that choosing one color all of the time would be the best strategy. If the hit color was black 67 times in a row...yeah, that would clearly look like a rigged system. On the other hand, if you are changing your guess around, it could just appear as "shitty luck" (answer A) when the hit color constantly does not match. The reason for this is that there is a second factor of randomness. When I'm picking the same color every time, the only random factor is the wheel result. When I'm flipping a coin, there are now two random factors. This, at least for me, gave the illusion of a reduced probability. And with some bad math and reasoning, I "justified" the illusion.

The actual answer is E. The probability of missing 67 times in a row is absurdly high, Jonathan, who's answer is posted, appears to assum the casino sees the result of your coin flip and they have figured out which way you pick based on that flip. His complete answer is here:
Answer from Jonathan: "The probability of losing 67 times in a row is one in 2^67, ie about 1 in 147 billion billion. So this is *extremely* unlikely to be bad luck. If the game is fair, flipping a coin is no worse than any other strategy - there's no pattern to pick up on. C is for idiots, D might make sense if you were always betting (say) red, but since your choice is random and there's no sensible way your coin toss can directly affect the wheel, if must be E, and the casino is seeing your bet, then manipulating the wheel (or, at least, it's far more likely that the casino is crooked than that you've lost fairly 67 times on the trot)."

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