Probability & The Bet (My Answer)

Here is my answer, there are no right or wrong answers. My answer is steered towards how I would make this intelligent bet with Mr Market(The Stranger).

1) A stranger walked up to me and say that he has 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?

Mr Market walked up to me and said that he has a fair coin. Imagine the process of flipping up to the 19th flip. After 5 flips, 5 Heads. I thought to myself, the odds of 5 Heads in a row is around 3%(0.5^5) since there is an equal chance of getting heads or tails. After 10 flips, 10 Heads(something must be wrong, my alertness increases). After the 19th flip, 19 Heads and 0 Tail(he must be kidding me!). Either this coin is biased(Mr Market lied to me that this coin is fair) or the odds of this probability happening is very rare! We live in not a perfect world, people lie, cheat, steal etc. The market is a brutal place.

My opinion would be that this coin is biased, more weight is placed at the side of tails so it always lands on tails and show heads on top. Since it is a biased coin, 99%(what if Mr Market do magic trick?) the next flip would be Heads.

Let us go back to purely numbers and data that this is indeed a fair coin(Mr Market is not lying). The flip is independent and probability is non-accumulative or you can say that there is no conditional probability attached. The next flip either heads or tails is 50%, is correct. Doesn't matter if it is 10 Heads in a row or 1 Billion heads in a row.

If you said that the probability of Tails occurring at the 20th flip is very high(eg. >50%). You are wrong. Past data of the flip cannot predict accurately the outcome of the next flip. It just shows you that a rare event just happened, which is consecutive 19 Heads in a row.

If you think carefully, there are a lot of lessons that we could apply in the market.

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?

I would use Kelly's formula(odds/edge) to bet. It is used to determine the optimal size of a series of bet when the odds are in your favor, in the long run. You are right no matter which case you choose because you do not know whether Mr Market is lying or not. Assuming 10k capital.

Case 1(not lying): 50% Head, 50% Tails, R/R 1:2

Your Odds:
(0.5 x 2) + (0.5 x -1) = 0.5

Odds/Edge:
0.5 / 2(divide by max win) = 25%

Bet 2.5K either heads or tails is the same.

Case 2(lying): 99% Head, 1% Tails, R/R 1:2

(0.99 x 2) + (0.01 x -1) = 1.97

1.97 / 2 = 98.5%

Bet 9.85K on heads.

However if I were to bet, I don't want to put 9.85k. I'm a conservative guy and don't want to risk almost all the money(call me a turtle gambler).

I bet on Heads by combining both scenarios adjusting probability to 100%:

(0.25 x 2) + (0.25 x -1) + (0.495 x 2) + (0.005 x -1) = 1.235

1.235 / 2 = 61.75%

Since I'm a turtle gambler and a value investor, with a 50% margin of safety, I would bet 3K(30%) of capital on Heads.

Kelly's formula also shows you how much confidence you have in the bet. In this case, my confidence level is 61.75%. If you think that the coin is fair, your confidence level would be 25%. In the stock market, most would bet if their confidence level is more than 70-80%. So if you said that you don't want to bet, you are right too. Like I said, there are no right or wrong answers, only bet with money that you can afford to lose.