Came up with this very lame question because I'm very bored haha... Posted it in Huatopedia forums and decided to post this Quiz/Question for my readers too! Hopefully we can learn from each other. If you could spend some time to think this through, you will learn something in the process. Good Luck!
1) A stranger walked up to me and say that he have a fair coin in his hand. Which means that the probability of flipping the coin to get either heads or tails is 50%. The stranger said that he will flip the coin 20 times and wants me to guess what is the result of the 20th flip. There were no tricks involved during flipping, no foul play seen.
After the 19th flip, the result was 19 heads and 0 tail. He asked me to guess the result of the next flip. What is the probability of getting either heads or tails on the 20th flip?
What is your answer and your reasoning?
2) The bet. The stranger said that if you want to bet, every $1 bet will give returns of 200%. If you lose, you lose all. Your R/R is 1:2. How many % of capital would you bet on this 20th flip to get optimal R/R returns, based on your probability answer given above?
Ps: R/R is Risk/Reward.
I will give my opinion in the next posting. =D
Edit: Spotted the mistake. The rest are correct. $1 bet will reward you $2, so total you will have $3. Sorry for the mistake.
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I use $1 to buy ToTo. Ha ha
LOL! Uncle88, in this case you can use $1 million to get another $1 million. :P Winnings will double your bet.
Probability 50%
Degree of certainty 99.99%
Bet tails
2.5X the amount till tails appear
Hey Cheng,
Ok, test book reply for question 1. The question is to be interpreted as “What is the probability of getting either heads or tails on the 20th flip, given that the first 19th flip are heads”. Since the probability of individual flips are independent from each other, the probability of getting a heads or tail of the 20th flip is the same as the n th flip, where n is any positive integer. Thus the probability is 0.5.
The above assumes that the coin is a fair die. From real world pt of view, it doesn’t seem like a fair die. I would say that the probability of getting heads seem higher than the tails, but the sample size is too small to determine the fairness of the coin. Given that the stranger comes up to me like this, I would not give him the benefit of doubt regarding the fairness of the coin.
Next…
For question 2, the expectation of the gains from playing the game is $0.50, based on 0.5 probability of getting heads or tails. However, using the statistical expectation is not a good estimate for this, since the number of trial is only 1 i.e. you’re only given one attempt.
Given this scenario, I’ll bet 100% of my capital, subjected to a cap of $2, which is what I can afford to lose for such a game in such a circumstance.
Very lame ans I know :)
LP, the probability of either heads or tails is 50%. However in gambling, there is another term called degree of certainty. Its like probability of probability.
Given any 1:2 payout for a coin flipping game, you can simply choose tails and double up every time it doesn't appear and reset your bet to $1 again whenever it appears. Chances are, the house will go broke after a while.
A high probability only increases the length of the streak. In this case, 19 x heads. At this point, the degree of certainty that a tail will appear is almost 99.9999%.
To put it into a simpler perspective, what is the probability of achieving 20 x heads if i flip a coin 20 x times?
Google for "The Fundamental Formula of Gambling" if you need more information.
”Let no one enter here who is ignorant of mathematics”
Hi iistery,
I read the whole thing about it and was humbled by it. If it is true, then the probability taught in school is both obsolete and impractical.
I tried to find out more info, but I couldn't. Is there more links to this theory?
Hi iisterry,
You are right, probability of 50% is correct. :D
However the degree of certainty which of 99.99% is something new to me. I would like to correct it and I rephrase in case others got it wrong that probability and "degree of certainty" is the same. They are different.
I do not doubt the fundamental formula of gambling that when you double your stakes at every loss until one day you get tails, you will be able to beat the house. This to me is degree of certainty of beating the house and is not probability. However as gamblers, we have limited capital and we are playing the greater fools game. By using this strategy, one would go bust very quickly.
Personally I would not recommend it, especially when dealing with the stock market. Either "you make or break" by risking all your capital or doubling it in itself is foolish. If you are lucky, confirm huat big! :P
In this case, kelly's formula would be more practical. Pardon me for being frank. :D
My answer is out, take a look.
Cheers!
Cheng :D
Hi LP & Cheng
Probability & Degree of Certainty are different. You've inadvertently explained it yourself with the following statement "It just shows you that a rare event just happened, which is consecutive 19 Heads in a row."
This is degree of certainty. The possibility that one can achieve 19 heads in a row is neigh to impossible. What about 25 heads in a row? Or 35? Hence you can see that the degree of certainty that a tails will appear is very high indeed. We don't talk about biased coins.
Other formulas that you use for determining the size of your bet is designed to minimise/maximise your loss/profit over a sustained course of action. Eg; X%.
No casino is a right state of mind would ever allow a game to be played with only 50% house edge. Hence you'll never see such a simple coin tossing game with no restrictions on bet.
No game is ever allowed in the casino unless it can be mathetically shown that they have a huge house edge. You need to understand this to know why you can never beat the casino at its own game.
Limited capital is the reason why ultimately gamblers lose. It is called "The Gambler's Ruin". You lose when eventually the losing streak outnumbers your capital.
You can't relate co-relate gambling with the stock market unless you're picking stocks by throwing darts. It is two entirely different matters.
Find out more at http://saliu.com/
But of course, we can all agree to disagree. It also explains why 1, 2, 3, 4, 5, 6 has never appeared in Toto despite having an equal probability as any other number.
Stock markets on the other hand have nothing to do with probability or gambling. Be it random walk or wave theories. As someone once said "I can calculate the movement of the stars, but not the madness of men."
”Let no one enter here who is ignorant of mathematics”
Hi iisterry,
Thanks for the reply. :D
I did mention that Probability and Degree of certainty are different.
On August 18, 1913, at the casino in Monte Carlo, black came up a record twenty-six times in succession [in roulette]. Rare events do happen, however how sure are you that the next flip would not be heads again? The probability is still 50%. Degree of certainly do not help us better understand the situation or make sensible decisions regarding risk taking.
In my opinion, the degree of certainty is a very stubborn theory, which is very one sided. If an immortal has been buying ToTo for 1 Million years and has never strike the top price before, would the chances of him winning in the next decade or 2 be higher than Uncle88? :P
http://www.fallacyfiles.org/gamblers.html
I believe that the stock market is like the biggest casino. Market makers win most of the time, they have the house edge. Every dollar that is won will be less than the dollar that is lost due to commissions and fees.
If one can understand gambling, probability risk taking and human psychology better, one can definitely improve their performance in the stock market.
Other than those above, I agree with the rest you said. :)
"I can calculate the movement of the stars, but not the madness of men." Einstein said this. The market is random. Hmmm... the more I think about it, degree of certainty is not very practical.
Regards,
Cheng :D
I must say that degree of certainty is definitely an not a correct theory (it's more like a gambler fallacy). If the coin is really true and fair, each event must be seen as independent.. :D
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