Happy Landlord as a classic national entertainment game, many people like to play a few rounds when they are bored. However, I also believe that everyone has had such an experience, that is, they obviously feel that they are playing well with landlords, but they can't escape the fate of losing all the happy beans.
Is it really too many masters of the happy fighting landlord? Is it really their bad luck? Today, Xiaobian will use the mathematical knowledge learned to study this problem.
To simplify the analysis, the following reasonable assumptions can be made:
1. The number of happy beans owned by the player is nk (here k is the unit, which means 1000);
2. The probability of the player's happy bean losing (becoming 0k) is T(n);
3. The number of Happy Beans that the player loses or wins in each round is 1k;
4. The probability of the player winning each round is p, the probability of losing is q, and the two meet p+q=1;
5. As long as there are happy beans, the player will continue to play until they lose or win all the happy beans of the system, which assumes that the number of happy beans of the system is mk (in fact, the system can have as many happy beans as it wants, and it is only recorded as mk here for convenient analysis).
The number of happy beans for the player is nk, if the next game wins the game, the number of happy beans will become (n+1)k, the probability is p, if the next game loses, the number of happy beans will become (n-1)k, the probability is q, so there is
According to this, you can get (friends who like mathematics can see the appendix) and the probability of the player losing
For the happy bucket landlord, the number of happy beans given to the player by the system every day is 6k, that is, n = 6, and the following figure shows the probability of the player losing T(n) with the number of happy beans in the system.
It can be seen that if the probability of a player winning or losing each game is 50%, then as the number of happy beans in the system increases, the probability of losing out is rapidly increasing. When the total number of happy beans in the system is 36k, the probability of players losing out is as high as 83%. In fact, the number of happy beans in the system can basically be regarded as infinite for the average player, and the probability of losing out is 1.
If the player's probability of winning each round drops to 45%, the probability of losing will further increase, and when the total number of happy beans in the system is 36k, the probability of the player losing out is already 1.
If a player is at a high level and has a 55% chance of winning each round, it can be found that as the number of happy beans in the system increases, the probability of losing out is only 30%. However, it should be noted that whether the landlord can win depends largely on the quality of the cards, and the player still has less time to get a good card, which ultimately keeps the probability of the player winning at 50% or less, and the probability of losing out is still 1 for the huge number of happy beans in the system.
The win rate of the landlord is 49%
The above analysis shows that for ordinary Happy Bucket landlord players, as long as they continue to play, the probability of losing all the happy beans is 1, which is why we always lose when we play Happy Bucket Landlord.
In fact, the above analysis process reproduces an interesting theorem in probability theory: in "fair" gambling, any gambler with a limited number of bets, as long as he gambles for a long time, will one day lose. This is the gambler's loss of light theorem.
The process of the happy landlord is in fact a gambler with limited capital and a banker with unlimited capital gambling, because the banker's funds are much higher than the gambler's, so he can always be invincible.
It's as if a gambler only has 1 yuan, while the dealer has 10,000 yuan, and it is very obvious who wins and who loses.
So even if both sides have a 50% chance of winning, the gamble is not fair from the start due to the difference in the stakes.
In addition, in the process of happy fighting landlords, if the number of happy beans of the player is larger, the number of matching players of the happy beans will also be more, and the system will also charge a corresponding handling fee for each round, so that the speed of the player losing the happy beans is further accelerated.
Therefore, everyone should be entertained in the process of playing happy fighting landlords, and don't care too much about the winning or losing of happy beans, because no matter how strong you are, you can't fight the probabilities in mathematics.
appendix
according to
Yes
Sum the left and right sides separately, yes
Obviously, T(0) = 1, because for the player who originally did not have happy beans, the probability of losing happy beans is naturally 1; and T(m) = 0, because all the happy beans in the winning system can not be played later, so the probability of losing is 0, thus
Eventually got
For the case of p=q=0.5, it can be obtained by finding the limit on the above equation
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Author: The little grapefruit that wants to change the world
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Does not represent the position of the Institute of Physics, Chinese Academy of Sciences
Source: Grapefruit Knowledge
Edit: Fun Superman