The binomial probability distribution is a discrete probability distribution that has many applications. It is associated with a multiple-step experiment (Anderson, Williams & Camm, 2018). A binomial experiment has the following properties: it must consist of a sequence of n identical trials. The experiment will consist of two outcomes; these outcomes are referred to as a success or failure. The probability is denoted with a p and these probabilities are of success. The four at-bat consists of n repeated trials, which can result in two possible outcomes; success outcome and failure outcome. Each trial is independent and the probability of success is p as constant for each trial, x is the random variable denoting the number of hits. A frequency chart is a two-column table lists each of the values in the data, and how many times it appears.
|
Number of hits (X) |
Frequency (F) |
Xi*Fi |
|
0 |
24 |
0 |
|
1 |
39 |
39 |
|
2 |
24 |
48 |
|
3 |
4 |
12 |
|
4 |
0 |
0 |
|
total |
91 |
99 |
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The numerical results describe the frequency of different number of hits with 4 at-bats for the baseball player. In this case, the number of times the player had 0 number of hits is 24, the player got 1 hit 39 times, 2 hits; 24 times, 3 hits; 4 times and never got to 4 hits. The total number of hits for the particular player at 4 at-bats is 91. The average number of hits is therefore given by
Probability distribution describes how probabilities are distributed over the values of the random variable. The reason we used the probability distribution is it will give us the probability associated with the event of P(x) successful shots respectively. Below, is the probability distribution obtained from the frequency distribution table. The table describes the probabilities of different number of hits to occur from the total number of hits. It is a probability distribution table as it the probability values are between 0 and 1 and the probabilities add up to one.
| Number of Hits | Probabilities |
| 0 | .2637 |
| 1 | .4286 |
| 2 | .2637 |
| 3 | .0440 |
| 4 | 0 |
| Total | 1 |
Figure 1 Scatter plot of the probability distribution
The batting average of any baseball player is the number of “hits” divided by the number of “at-bats”. When we use the frequency distribution on Part 3 of this problem, the player’s batting average can be calculated by first finding the total number of “at-bats” as well as the total number of hits (r). The batting average therefore is given as
Therefore, the batting average can be given as:
The binomial probability distribution of four trials can using the batting average of is shown below.
|
Number of hits (k) |
P(X=k) |
|
0 |
0.2809 |
|
1 |
0.4198 |
|
2 |
0.2352 |
|
3 |
0.0586 |
|
4 |
0.0055 |
The probability of mean number of successes is the number of hits made taking into account the total number of “at-bats” and its is The scatter plot for the binomial distribution from excel is shown below.
Conclusion
The binomial distribution and the probability distribution of four at-bats differ. This scenario is further illustrated by the shape of the scatter plots in both cases. The difference emanates from the fact that binomial distribution is discrete and not continuous and is therefore not possible to find data between any two data values while the probability distribution of four at bats accommodates continuous variables. The mean in part 4 is 1.08791 while the mean of part 6 is 0.27198. The differences come about as in part 4, the frequency represents the number of hits after 4 at-bars while in part 6, number of hits is divided by the total number of at-bats. The total number of “at-bats” is given by multiplying the total frequency in part 4 by 4.
Reference
Anderson, D. R., Sweeney, D. J., Williams, T. A., Camm, J. D., & Cochran, J. J. (2018). Essentials of statistics for business & economics . Boston, MA: Cengage Learning.
Descriptive and Inferential Statistics. (n.d.). Retrieved from https://statistics.laerd.com/statistical-guides/descriptive-inferential-statistics.php
Z-Score: Definition, Formula and Calculation. (n.d.). Retrieved from https://www.statisticshowto.datasciencecentral.com/probability-and-statistics/z-score/
Skellam, J. G. (1948). A probability distribution derived from the binomial distribution by regarding the probability of success as variable between the sets of trials. Journal of the Royal Statistical Society. Series B (Methodological) , 10 (2), 257-261.
