Can three cobblers really be better than Zhuge Liang?

"Three cobblers are better than Zhuge Liang" is a Chinese proverb, which comes from "Romance of the Three Kingdoms".

In order to implement the "borrowing arrows with straw boats" plan, Zhuge Liang ordered three of his subordinates to put straw targets on straw boats and then cover them with cloth to disguise them as soldiers. However, his subordinates felt that this arrangement was not appropriate and would reveal flaws, so they placed straw men on the bows of each boat and put on leather clothes and leather hats. Later, Cao's army really fell into the trap.

This sentence means that the wisdom of three ordinary people combined may surpass that of one very smart person. It emphasizes the importance of teamwork. Even if the individual ability is limited, through cooperation, greater strength can be exerted. Zhuge Liang is a famous wise man and strategist in the history of our country, and here he is regarded as the embodiment of outstanding wisdom.

So the question is, can three people of average ability really surpass one person of outstanding ability by working together? If you are one of the three "cobblers", how can you find a way to surpass "Zhuge Liang"?

Let's look at the simplest mathematical model: suppose there are three people with average abilities, named A, B, and C, and the probability of them solving a certain problem independently is 0.3, 0.4, and 0.5 respectively; there is another person with outstanding abilities, named D, and the probability of him solving this problem is 0.8. Now A, B, and C form a team. To simplify the model, they still solve the problem independently. The probability that none of them can solve the problem is (1-0.3)×(1-0.4)×(1-0.5)=0.21. Therefore, the probability that at least one person in this team can solve the problem is 1-0.21=0.79.

A magical result appeared. After three people formed a team, the probability that at least one person would solve the problem was very close to 0.8.

In order to surpass 0.8, we can try the following two methods: First, try to replace a teammate with a slightly stronger ability, such as replacing A with a person whose probability of solving the problem alone is 0.4, then the final result will become 1-(1-0.4)×(1-0.4)×(1-0.5)=0.82; the second method is to let more teammates join, such as adding a teammate with a problem-solving ability of 0.3, then the team's probability of solving the problem will become 1-(1-0.3)×(1-0.4)×(1-0.5)×(1-0.3)=0.853. You can try it. If the team grows to 10 or even 100 people, the probability of solving the problem is much higher than 0.8, and even close to 1.

The above is a simplified mathematical model. In reality, different people have different skills and knowledge. Cooperation can often inspire new ideas and increase the ways and methods to solve problems, thereby increasing the probability of solving problems.

“Three cobblers are better than one” emphasizes the importance of cooperation, diversity and cumulative effect in problem solving. As the saying goes, many hands make light work. On the road to success, we can fully believe in the power of the team.

(The author Chi Hongmei is a key teacher of the university mathematics teaching academic team of the School of Information, Huazhong Agricultural University)