Is there any hope of qualifying for the last six victories of the national football team?

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Source: Zhihu.

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The third stage of the Asian World Cup qualifiers: 12 qualifying rules for top-level events. 12 the team is divided into two groups, and then the home and away games are played in double round robin. The first two teams directly advanced to the finals of the World Cup in Russia, and the two teams decided the play-off places with the third place to compete with other continents for another World Cup place.

After four rounds in China's Group A, the points are as follows:

In other words, if China wants to qualify, it must enter the top two or the third place in the group, and the third place in the other group will compete for the last place.

Let's see the probability that China will enter the top two in the next six rounds.

As we know, there are only three situations in a football match: win, lose and draw, so suppose that the probability of China winning in each match is 1/3 (of course, we know that this is impossible, because the probability of China winning is much lower than 1/3, right? But for the convenience of calculation, we assume that), and each game is an independent event.

The probability of China winning the next six games is 1/3 to the sixth power = 1/729. If you win all six games, you can accumulate 19 points. Next, let's see how likely it is to rank in the top two of the group with 19.

To be the first in the group, the points of other teams must be 18.

The probability that Iran scored 18 is 5/243 (the sum of the probabilities that Iran scored 8, 7, 6, 5, 4, 3, 2, 1 and 0 in the last six rounds).

Similarly, the probability of Uzbekistan 18 is 17/729.

The probability of Korea 18 is 20/729.

Syria and Qatar don't count! ! !

Then calculate the probability of China team's second place in the group, and then calculate the probability of team's third place multiplied by 1/2 (assuming that group B beats the third place).

It's so annoying, as a person who hasn't studied mathematics for many years. . .

Finally, adding up the probabilities of meeting the above three conditions at the same time, China team may qualify, which is probably millions of figures, which may be slightly higher than the probability of winning the first prize in the lottery.