A gambling game, toss a coin, if it is heads, you give me 100 yuan, if it is tails, I will give you 150 yuan. Do you want to play?

For you, the mathematical expectation of this gambling game is150 * 50%-100 * 50% = 25 yuan (average expected return), which is quite cost-effective, but many people are unwilling to gamble, because they hate losses, and there is still the risk of losing100 yuan.

Want to bet 10 thousand, still want to play? Of course. The law of large numbers tells us that when there are enough data, its frequency will be close to its theoretical probability. You may lose if you gamble one or two, but 10,000 will almost even out all the fluctuations, and your expected return is 25 *10000 = 250,000.

Real life cannot be so simplified. The utility of 100 yuan is different for people with an annual income of100,000. In real life, no one will bet you 10,000 yuan. Real life is immeasurable. That boy looks reliable and handsome. Should I give up all my body and mind? English is so difficult that I can't remember the words. Should I take a class?

This thought experiment just tells you that life is not a game, but a repetitive game. Before you die after a fall, as long as your mathematical expectation is positive, you should keep trying, just as Daniel Kahneman suggested in Thinking, Fast and Slow: No one really gambles with you for 10,000 times in life, but you can consider all the probability choices in life as a system. As long as you choose to gamble every time your math expectation is positive, you will win in the long run.

You should think like a businessman, bet according to the probability, don't worry too much about the sunk cost, don't care about one day, you have to spend your whole life.

(2)

There is a gambling game in which you win 2 and lose 1. If you flip a coin to bet, the heads will double and the tails will lose the principal. Your total gambling amount is 100 yuan, and you can invest any amount in each bet. How do you bet?

If you like adventure, you can choose to play big, 100 yuan is all on it. If you lose, you lose everything. If you win, you make a lot of money! If you are conservative, you will be cautious. Take your time and play only one dollar at a time. Only bet 1 yuan at a time, winning 2 yuan head-on and losing 1 yuan head-on. But after playing 20 times, I feel that I haven't won as much as 20 yuan once after working hard for a long time, which is so humbled.

100 is too much and 1 is too little. How much should I bet? Kelly criterion tells you the answer:

f* =(bp-q)/ b

In the formula, the meaning of each parameter is:

F* = capital ratio to be bet.

P = the probability of winning (that is, the probability of turning the head)

Q = the probability of failure, that is, 1-p (that is, the probability of coin tails).

B = odds, equal to expected profit-possible loss (that is, profit-loss ratio)

The molecule bp-q in the formula stands for "winning face" and is called "expected value" in mathematics. Assuming that the probability of winning is 75%, the probability of rewarding is 80%, the probability of losing is 25%, and the penalty is 150, then its mathematical expectation is 75%*80-25%* 150=22.5.

Substituting into the formula, you should invest (bp-q) ÷ b = (2 * 50%-50%) ÷ 2 = 25% every time you flip a coin, that is, you should invest 25% of the total gambling money every time, 25 yuan. Kelly's rule tells us that only by choosing the best betting ratio can we get the highest income for a long time and avoid losing our ass completely.

The formula is beautiful, but the meaning is difficult to understand. When do you want to go all in? What if you win more than twice? How about 10 times, 100 times and 1000 times? Calculate, even if it is 1000 times, your maximum bet ratio is only 0.4995. As long as the probability remains the same, don't go all in, you will always lose.

(3)

In the casino, the banker's advantage is the smallest, 2 1 point, and the banker only wins the player by 0.4%, about 50.2%, and the player's winning rate is about 49.8%, so you know why you lose in the long run.

You believe in luck, and casinos believe in probability. Most people want to beat the casino by luck, but they don't know that behind the casino is the probability calculation that you can't shake. Even if I tell you the rules of counting cards, most people's ability to calculate single digits is not enough for online use.

Winning 2 1 is a movie about a group of MIT students playing casino. Its true prototype comes from Edward? Professor Thorpe.

There are many changes in the rule of 2 1 minute, but simply speaking, the dealer takes turns to deal cards. Whoever takes the lead in exceeding 2 1 minute will explode and the rest will win.

When the number of points is less than 17, the dealer must continue to ask for cards, and the player can choose whether to take them or not. At this point, if all the cards appeared before are small cards, then there will be more big cards in the remaining pile, and the dealer will be more likely to explode. Players should increase their chips at this time.

Thorpe figured out a set of card counting skills, guessed the number of cards in the dealer's hand with a probability of more than 50%, and used these probability points and Kelly formula to optimize his best bet ratio every time and continue to bet. He beat the casino.

To put it simply, Thorpe follows the principle that heavy cards are played when the odds are high and small cards are played when the odds are low.

The use of expected returns/odds is aimed at people in casinos. For ordinary people, its general meaning is to use advantages/disadvantages. A game, see if you have an advantage. If the expected return is positive, you can continue to play, bet rationally, and keep the possibility of sitting on the table and playing until your opponent dies. If you have a big disadvantage and the expected return is negative, you must not play because you are a fish.

(4)

Life is long, not a game, but a series of events. Find out what you think is right and stick to it, so that others can no longer ignore and catch up with your accumulated advantages. Find a long slope with slight wet snow on it, circle your little snowball and push it down.

So what is worth all-in?

I think learning is always on the list. Not only in investment, but also in leverage, as Soros once said: for high-certainty transactions, we must take the throat straight, not only invest our lives, but also borrow money to gamble. That's why Paulson Fund made a huge profit of $4 billion by shorting American subprime loans in 2007.

In places where you are uncertain, always be cautious and optimistic, and don't be left behind. No matter how big the game is, it has nothing to do with you. As Roosevelt once said, "Honor belongs to those who are really in the arena."

Finally, to borrow a sentence, "faith and patience can help you overcome accidental disappointment and wait for inevitable good luck."

Refer to the column of Wan Weigang, Wang Shuo and Lao Yu WeChat WeChat official account's "Lonely Brain".