An unbiased coin is going to be tossed 100 times and no betting is to take place until after the first 10 tosses. The first 10 tosses come up heads. The bet is: will heads maintain a 10 point lead at the 100th toss?
There is a 50% chance that there will be a 10 point lead after the 100th toss.
To save time, just do one heads or tails coin flip.
And once again, just like roulette, craps and coin flips, what happened on the previous roll, toss, or flip has no influence on the next roll, toss or flip.
Since this is a completely fair gamble, there is no sense taking the bet. You have no edge.
Think about it a little.
Math Boy
stays at ten or increases over the next 90 flips...
It is a sampling problem we've been presented with.
After 10 straight flips with the same outcome 10 I would begin to doubt that was a fair flipping of a fair coin...
I believe the odds of getting 10 in row would be 1 divided by 2raised to the ninth power... 1/512...it could happen in a fair test, but not likely...
so, I would bet a dollar that the over the next 90 flips the lead would be maintained, BUT because I suspect an unfair test, I would suspect the outcome is controlled, perhaps to con me into betting a large amount.
Yeah, I guess I totally overlooked the idea that you will win if the 90 remaining tosses go 45/45 in which case you would win, thus you have a .9% advantage on that bet
You have an edge if it is a fair coin. If it is an unfair coin, it is most likely biased towards your bet.
I'll bet people on coin flips sometimes for kicks. Some people won't cover the coin as they catch it and I can call it. At worst that bet is 50/50.
Math Boy
If I understand the rules, I win if there are 55 or more heads by the end, so that 45 or fewer tails. In other words, I win if the next 90 tosses are 45 or more heads.
My edge is the Probability that the next 90 has exactly 45 flips. This is Combin(90,45)/2^90. According to my calculations, this is about 8.4%. This is an even-money, so this will call for a wager someplace in the area of 30 �units�, if your Kelly Equivalent Bank is 400 "units".
If you having trouble seeing that the edge is 8.4%, note that probability of getting more than 45 heads if 50%-8.4/2% =45.8% The probability of getting fewer than 45 heads is also 45.8%, which is the probability of losing The probability that the number of heads is >=45 is 45.8%+8.4% = 54.2%. The probability of winning minus Porb of losing is 54.2-45.8 = 8.4%
that ripped Uston off in his book "The BigPlayer"??? I think that is where I read the story about losing a bundle to a coin flipper, although it might have been in an interview somewhere also...
:)
With all due respect, and hopefully you'll make allowances for my use of your math language.
If you look at the 90 remaining tosses in isolation, how can you possibly have an edge?
If you include the previous 10 tosses, you would have to conclude that one is seeing an unusual distribution of heads and is MOST LIKELY just variance in this even game. Each of the first 10 tosses, also had the probability of coming out even, yet they didn't.
I would think that the probability that the difference is variance, is a higher percentage than the 50% chance of an even split on the remaining 90 tosses.
Shouldn't the higher probability rule and determine the bet?
Whether you flip a coin 10 times, 50 times, or 90 times, the average event is 50/50. If heads has a 10 point lead or if you did a rain dance while standing on your head makes no difference. North Wind, what if I flipped a coin 10 times and never told you the outcome. What would you bet on next? Wouldn't you feel just as safe betting one way or another? I don't know how else to say it. To assume that just because heads came up 10 times in a row that tails is MORE LIKELY to come up now is absolutely dumb. Even a 1st grader could tell you that. Don't you think that in the year 2005, probability and statistics has explored every possible angle of theory and is now declared fact? Varience is just like luck, you can only determine that, based on what already happened and cannot predict what is going to happen next.
MathProf is right. The key point is that when the outcome is exactly 45 to 45. The question is: Maintain the lead. There are different outcome such as Win:Lose 35:55, 47:43, 50:40, etc. So in the situation of getting exactly the 45:45 chance (Combin(90,45)/2^90) fits into Maintain the lead side. You really have an advantage of betting it on Maintain the lead side. OTOH, If you change the question from the next 90 tosses to the next 91 tosses, then everything is equal. You have 45:46 and 46:45 but have no chance of having exactly even situation. Betting on Main the lead side is the same as against it.
Math Prof doesn't really hinge his answer on the first 10 flips. The "first 10 flips" part is really just a ruse to distract you from the real advantage in the wager. The key to the wager hinges on the "maintains the lead" part.
In an ordinary coin-flip wager, no one wins when a coin is flipped 90 times and the result ends up 45 Heads and 45 Tails, but the wager being discussed here is slightly different. In the "maintains the lead" scenario, you win if you wager that the lead will be maintained and the outcome of the 90 coin-flips turns out to be 45-45. That's a nice advantage to have. Math Prof calculated the chance of a 45-45 tie occurring.
You win if the outcome is 90 Heads and 0 Tails. On the other hand, you lose if the outcome is 0 Heads and 90 Tails. Both outcomes are equally likely when using a fair coin. You win if the outcome is 89 Heads and 1 Tail, but you lose if the outcome is 1 Head and 89 Tails. You can go on like that, thumbing through all all possible outcomes until you've included 46 Heads-44 Tails paired up against an outcome of 44 Heads-46 Tails. The lone outcome remaining is 45 Heads - 45 Tails. Since that outcome is a winner (it maintains the lead), it makes the wager profitable.
Titan5 pointed out that it's the even number of flips (90) that makes the wager profitable. With an odd number of flips, there is no lone outcome with which you can gain the advantage. All the winning outcomes are exactly matched up with losing outcomes.
a tongue-in-cheek question of course. :)
But the similarity between your story and his was too good to pass up...
:)
