Probability Distribution Theory: The Key to Understanding Data

Introduction 

Probability distribution theory is an important concept in our daily lives. The knowledge of probability distribution theory is widely applicable in statistics, data analysis, planning for weather, sports strategies, gambling, insurance, and so forth ( Tuckwell, 2018 ). For instance, insurance companies utilize the concept of probability in calculation of premium rates when issuing a life insurance. Actuaries normally determine the level of risk being assured through calculation of the probability that the event will occur ( Tuckwell, 2018 ). A higher possibility of an event occurring implies the assured will have to pay higher premiums. For example, an insurance company is issuing health insurance policy to two individuals, a smoker and non-smoker. According to statistics, smoking is harmful to health thus increases risk of illness. The probability of a smoker developing health complication is higher compared to that of a non-smoker. The insurance company will therefore charge the smoker more premium rates as compared to non-smoker ( Tuckwell, 2018 ). 

The sports industry is another field where probability concept is greatly applied, from sports gambling to sports strategies. Probability theory can be used to analyze the performance of a player ( Tuckwell, 2018 ). For instance in baseball, the performance of a player is determined by the number of hits per at-bats. A manager will determine how good a player is by evaluating the batting average of each player. This paper demonstrates how the probability theory can be applied to determine average batting of a player. It also evaluates if binomial probability distribution can be used in baseball game experiment by finding whether the batting experiment satisfies all the conditions of binomial distribution theory. 

Question 1: Explain why the four at-bats is a binomial experiment 

An experiment is said to be a binomial experiment if it has the characteristics; 

The experiment has fixed number of trials, and each trial is independent of each other ( Ayyub & McCuen, 2016 ). 

Each trail must have two mutually exclusive outcomes; success and failure ( Ayyub & McCuen, 2016 ). 

The random variable (x) value must be discrete ( Ayyub & McCuen, 2016 ). 

In this batting game, the player has fixed number of trials; he has exactly 4 at-bats. The results of each at-bat (number of hits) are a discrete random variable and independent of each other. Also, there are two mutually exclusive outcomes; a player getting a hit (Success) or he does not get a hit (Failure). This experiment satisfies all the conditions of a binomial distribution. Thus the batting game experiment is a binomial experiment. 

Question 2: Construct a Frequency Distribution for the Number of Hits 

Frequency Distribution Table 

Frequency Distribution Table
Number of Hits (xi)	Frequency(fi)
0	34
1	36
2	15
3	1
4	0
Total	86

Question 3: Compute Mean Number of Hits 

Mean

Frequency Distribution Table
Number of Hits (xi)	Frequency(fi)
0	34
1	36
2	15
3	1
4	0
Total	86

Therefore, 

Mean =

=0.802326 

Because decimal value of hits is not relevant, the value 0.802326 is rounded to the nearest value, 1. The mean number of hits of hits is 1. This numerical value implies that after every 4 trials, the player hits only once. 

Question 4: Construct Probability Distribution 

Using the frequency distribution, it is possible to determine the probability of the number of hits the player can get. It is the ratio of frequency of number of hits to the total frequency. 

Let x­ i be the number of hits, the probability of the random variable X is equal to x­ i is given by; 

P (X=x) = 

P(X=0) = =0.395 

P (X=1) =  =0.419 

P (X=2) =  = 0.174 

P (X=3) =  = 0.012 

P (X=4) = = 0 

Therefore, the probability distribution table is given by; 

Batting Probability Distribution
x	f(x)
0	0.395
1	0.419
2	0.174
3	0.012
4	0

For a distribution to be considered a probability distribution, the probability of each random variable must range between 0 and 1, and the total probability must equal to one ( Ayyub & McCuen, 2016 ) . For the above distribution of number of hits, each probability value ranges between 0 and 1. Also, the total probability of the distribution is 1; 0.395 + 0.419 + 0.174 + 0.012 = 1. Therefore the distribution of the number of hits is a probability distribution. 

Probability Distribution Scatter Plot 

Question 2: Calculate the Player’s Batting Average for Four At-bats 

The formula for batting average is given as; 

Batting Average =

Batting Average =

= 0.20 

Question 6: Construct Binomial Probability Distribution 

Using the batting average as the probability of success, this batting game is a binomial distribution with parameters number of trials (n) = 4, and probability of success (p) =0.20. In excel, the probability of a random variable X takes the value k, that is p (X=k) is given by the formula; 

P(X=k) = BINOM.DIST (k, n, p, False) 

Using this formula, the following binomial distribution was constructed. 



Binomial Distribution, n=4, p=0.20
Number of hits	Probability
x	p(X=x)
0	0.4096
1	0.4096
2	0.1536
3	0.0256
4	0.0016

Question 7: Mean of Binomial Distribution 

Mean of a binomial distribution is given by, mean= n (number of trials) times p (probability of success). 

For this batting experiment, n= 4 trials and p=0.2. Thus; 

Mean = np = 4 *0.2 = 0.8. Question 8: Scatter Plot for Binomial Distribution 

Using excel, the following scatter plot was constructed. 

Question 9: Compare Batting and Binomial Distribution 

In the batting distribution, the distribution of the random variables (0, 1, 2, 3, and 4) is given by (0.395, 0.419, 0.174, and 0.012). The distribution of random variables (0, 1, 2, 3, and 4) in the binomial probability distribution is given by, (0.4096, 0.4096, 0.1536, 0.01256, and 0.0016). Checking at the skewness of both distributions; 

As it can be seen, both distributions have a similar trend; they are both skewed to the left. Another similarity between the two distributions is the mean. Both distributions have a mean value of 0.80. However, there is a slight difference in the probability values. For instance, the random variable 4 is assigned a probability value of 0.0016 in the binomial distribution, while in batting distribution it has a value. This difference is due to the fact that binomial distribution assumes that under each trial, two mutually exclusive outcomes must occur, success and failure. Therefore, probability values are distributed across all the random variables, while in the batting distributions, probability values are assigned according to the real life events recorded. 

Conclusion 

This study has shown how probability theory can be applied in sports. The managers can use probability theory to determine the average performance of a player. For instance, baseball player with impressive batting average per game is regarded as a good player. This probability study reveals that batting experiment assumes the model of binomial probability distribution. The comparison between batting game distribution and binomial distribution reveals that the calculated batting average using both distributions is the same. Batting distribution meets all the requirements of a binomial distribution; it has fixed number of trials, the trials are independent of each other, and each trial has two mutually exclusive outcomes (success and failure). For this study, it is reasonable to conclude that batting game is another example of binomial distribution theory. 

References 

Ayyub, B. M., & McCuen, R. H. (2016).  Probability, statistics, and reliability for engineers and scientists . CRC press. 

Tuckwell, H. C. (2018).  Elementary applications of probability theory . Routledge.