Chao Huang

Chao Huang

Ph.D. student in M.E. at UTokyo.

Statistics [01]: Probability vs Statistics

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Probability and statistics are two closely related fields in mathematics which concern themselves with analyzing the relative frequency of events.. Still, there are fundamental differences in the way they see the world.

To better understand this, let’s start from an example.

Example

Suppose A and B were palying a best three out of five game with 100 dollars capital each, and the winner would win all the 200 dollars. However, due to certain reasons, the game had to be canceled when A had a 2:1 lead. Here comes the question: What is the fairest way to allocate the 200 dollars?

Solution 1: As there is no final result, A and B should each obtain their 100 dollars.

Solution 2: As A has a 2:1 lead, A should obtain all the 200 dollars.

Solution 3: As A has a 2:1 lead, A should obtain two thirds of the 200 dollars.

Solution 4: Suppose the probability of A and B winning the game are p_1 and p_2, respectively. A should obtain p_1/(p_1 + p_2) of the 200 dollars.

Considering the four possible solutions above: solution 1 may be unfair to A; solution 2 may be unfair to B; solution 3 may be unfair to both A and B; and solution 4 seems to be a reasonable way of allocating the money. However, here arises another question: how can we know the probabilities p_1 and p_2? This is where the statistics will take effect. Given the historical data of the game between A and B, the job of statistics is to analyze the frequncies of A and B winning the game respectively, which can then be used as approximations of p_1 and p_2 respectively.

Now suppose the probability of A winning each game is p and there is no draw, then p_1 = 1 - (1 - p)^2 = 2p - p^2, and p_2 = (1 - p)^2. Let p_1 = p_2, we have p = 1 - \sqrt{2}/2. This indicates that when p = 1 - \sqrt{2}/2, the probabilities of A and B finally winning the game are the same; when p > 1 - \sqrt{2}/2, the probability of A finally winning the game will be greater than B; and them p < 1 - \sqrt{2}/2, the probability of B finally winning the game will be greater than A. This is exactly the job of probability, given the probabilities obtained from statistics, it deals with predicting the likelihood of future events.

From the above example, we may fairly say that probability studies the consequences of mathematical definitions, and statistics tries to make sense of observations in the real world. In another words, the essence of probability is from rules to results, whereas the essence of statistics is from results to rules.

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