Answer

A random variable X follows a binomial distribution if its probability space consists of probabilities of two possible outcomes of a phenomenon – success or failure (DeGroot & Schervish, 2012). A binomial experiment describes experiment consisting of n repeated trials, with outcomes that can be defined as either success or failure. Further, the distribution is defined by two significant properties, number of trials (n) and probability of success (p). This paper is an analytical approach of the fit of binomial distribution in a base-ball player’s at-bat results from results obtained from his baseball games.  

This is a binomial experiment because in every at bat, there are two possible outcomes in the sample space – either hit or out. In this case, success is defined by the player hitting the base, an out defines failure.

Table 1: Frequency table of the number of hits in games.

The mean of the distribution is 1.088. This means that it is expected that the player will be getting above 1 at-bat in each game averagely.

This is a probability distribution because it covers the entire probability space, hence can be defined along the underlying sample space whose probability P(S)=1. Moreover, since the model here is a set of phenomenal outcomes (0, 1, 2, 3, 4), making it a discrete probability distribution (Chen, 2010). The specific probabilities developed from frequencies provide the probabilities of outcomes/occurrences of the phenomenon (at-bats). In this scenario of fixed four at bats, the probability distribution in case that the player has the highest chance of hitting one at-bat than he does all the other possible outcomes. This defines the probability space based on the sampled games with number of hits with four at-bats. 

Fig 1: Probability Distribution plot.

The batting average for four bats is

Average =

=99/(4×91)

=0.272

In calculation of the binomial probability distribution, the probability for 4 bats is fixed at 0.272

Therefore, the binomial distribution is determined by parameters n = 4 and p = 0.272.

Fig 2: Scatter of the Binomial Probability Distribution

Mean = np

=4*0.272

=1.088

Though the means are relatively equal, the probability distribution and the binomial distribution differ significantly, based on the player’s outcome at four at bats. For 0, 3 and 4 hits, the player’s outcome (in the probability distribution) is higher than the expected (in the binomial distribution). The player also had more 1-hit and 2-hit than expected. This does not mean that binomial distribution formula is a poor fit for the player’s at-bat results, as also argued out by Abelson (2012). 

Therefore, though binomial distribution shows significant variations with the probability distribution of the observed data of the player, it fits well in estimating the player’s bat results.