The term probability is frequently used in everyday life such as probability of rain, probability of passing the entrance examination, and probability of winning lottery. It is also widely known that probability is a concept that takes a value ranging from 0 to 1, and the closer the value is to 1, the higher the probability will be. This article will provide familiar themes and introduce the way of thinking in probability theory.
Mean used in probability theory
Mean is a representative value of the distribution characteristics. Arithmetic mean is used to calculate the value for mean scores, mean age, and mean income etc. When we calculate arithmetic mean, we give all factors the same weight. For example, the mean age of 10 people can be found out by adding the ages of the 10 people together and dividing it by 10.
On the other hand, mean which takes the weight of the factors into consideration is called weighted mean. Let us suppose the case where your grade is decided on the mean score of four examinations. From the first to the fourth examination, you get the scores of 60, 55, 80 and 85 respectively. At this stage, the mean of the four examinations (arithmetic mean) is 70. Since the range for questions expands for each following test, we decide to give each test a different weight. Let us give the first examination a weight of 0.1, the second 0.2, the third 0.3, and the fourth 0.4. Then, the weighted mean for the examination scores is
60 × 0.1 + 55 × 0.2 + 80 × 0.3 + 85 × 0.4 =75.
Here, in comparison to the arithmetic mean for the four examinations of 70, the weighted mean of 75 is higher because you get higher scores in the third and fourth examinations where the weight is heavier.
Mean used in probability theory is the weighted mean of the quantity determined by trial results, which we call expectation. When we calculate expectations, probability is used as weight. In the example of the four examinations, we can assume that the first to the fourth examinations are assigned respective probability of 0.1, 0.2, 0.3, and 0.4. Here, the total of the probability should become 1. When all the weight is equal (in the above case all weight is 0.25), the expectation is consistent with the arithmetic mean.
Quantity indicating dispersed degree
Mean is useful when showing the single value of distribution characteristics, but sometimes we need more detailed information.
For example, when you take a bus, it is important whether a bus is running on time according to the timetable or not. Here, the expected time of arrival stated on the timetable is the mean time. However, depending on traffic conditions, the arrival time is dispersed from the mean time. The convenience of uses depends on whether a bus arrives within three minutes either side of the expected arrival time, or 10 minutes either side. The quantity indicating this type of dispersed degree is called variance. So, the larger variance is, the larger dispersion is from mean.
Here we will look at an example of asset management. For a total of 1,000 thousand yen managed for one year, we will compare the following three financial products.
(1) Return of 1,020 thousand yen after one year in a fixed term deposit.
(2) 0.5 probability of returning 1,030 thousand yen, and 0.5 probability of returning 1,010 thousand yen in bond investments.
(3) 0.8 probability of returning 1,060 thousand yen, and 0.2 probability of returning 860 thousand yen on the stock market.
The expected amount of return for all three products after one year is 1,020 thousand yen. For example, the calculation formula for case (3) is
1060 × 0.8 + 860 × 0.2 = 1020 ( thousand yen)
The difference is the degree of dispersion. In case of (1), we can expect a certain return of 1,020 thousand yen after a year. In case of (2), there is dispersion of 10 thousand yen from 1,020 thousand yen, and the degree of dispersion from the expectation becomes even larger in case of (3). Furthermore, in case of (3), there is a chance of receiving 1,060 thousand yen with probability 0.8, but there is also a chance of the 1,000 thousand capital taking a great drop in value to 860 thousand with probability 0.2. Thus, the case of (3) is the riskiest investment of the three. In financial engineering, variance is used as a risk indicator in various numerical formulae.
Next I will explain deviation value, which converts scores using mean and variance. If you take many examinations, you will see that the mean scores and distributions change every time depending on the difficulty of the questions. Deviation value is used there as an indicator of your own ranking among all examinees. I will omit specific formulae, but in converting scores into deviation value, the mean score is first given a deviation value of 50. Then it is assumed that the scores are distributed in a symmetrical bell curve (this is called normal distribution), centered around the mean score (the deviation value of 50). Through conversion using mean and variance, 68.27% of all examinees will be included between the deviation values of 40 and 60. The top 15.87% will have a deviation value above 60, and 2.28% above 70. 99.73% of all examinees, in other words almost all examinees, will lie in the deviation range of 20 to 80. But, there is the possibility that, if an examinee’s score is extremely high compared to the mean and other examinees’ scores, his deviation value will exceed 100, conversely, if his score is extremely low, he will have a minus deviation value.
Law of large numbers
One important theorem in probability theory is the law of large numbers. This is the law where, as the number of trials increases, the arithmetic mean of the quantity determined by trial results will theoretically approach the expectation. Here, we assume that result from each trial does not influence each other.
For example, let us think the ratio of tails appearing when we toss a coin. Here, the theoretical expectation of the ratio is 0.5. Suppose an experiment of 10 coin tosses was taken and the result was seven heads and three tails. That is to say, the ratio of tails gained in the trial result is 0.3. However, if the number of coin tosses increases to 100 times, or 1,000 times, the ratio of tails will approach the theoretical expectation of 0.5.
Let us apply the law of large numbers to winning lottery amounts. Imagine a lottery where one ticket costs 200 yen and the first prize is 100 million yen. Now let us think about how much prize money someone could win if he had 50 million yen on hand and spent it all on lottery tickets. First we find the expectation of the winning lottery amounts. By calculating from the total number of tickets sold and total prize money, the winning amount expectation for a 200 yen lottery ticket is usually about half of that. However, this is the theoretical expectation, and in fact there are some lucky people who strike the 100 million yen jackpot with a single 200 yen ticket. Because the variance in the case of a lottery is extremely large, compared to a coin-toss, a huge number of trials need to be taken for the law of large numbers to be realized. So, when one purchases a huge number of tickets, the amount of his winnings becomes closer to half of the money spent. Therefore, if someone spends 50 million yen on lottery tickets, he may suffer great losses because the winning amount will be close to 25 million yen, which is expectation.
I have given a simple introduction to the basic thinking in probability theory. When mentioning probability theory, you may have a strong image of counting up a number of all cases. But probability theory learnt at university is quite different from that of high school as it deals with the concept of continuous probability distribution. To calculate probability for continuous distribution, we use integral calculus. Normal distribution, which appeared in the explanation about deviation value, is an example representative of continuous probability distribution.
Here I have taken up familiar and simple subjects, but probability theory in recent years has been applied in various fields such as economics, engineering, and so on. Through this essay, I would be pleased if you gained, even if just a little, an interest in probability theory.
Professor of Probability Theory, Faculty of Economics, Chuo University