Probability Analysis of Customer Purchases at a Television Store

As a television store owner, it is important to understand customer behavior and make informed decisions on inventory management and sales strategies. In this blog post, we will analyze the probability of customer purchases based on a given scenario.

A television store owner figures that 50 percent of the customers entering his store will purchase an ordinary television set, 20 percent will purchase a color television set, and 30 percent will just be browsing. If five customers enter his store on a certain day, what is the probability that two customers purchase color sets, one customer purchases an ordinary set, and two customers purchase nothing? 

To solve this problem, we will use the multinomial distribution, which is a generalization of the binomial distribution that describes the probability of observing a set of counts in multiple categories. In this case, the categories are color television sets, ordinary television sets, and browsing customers.

Let's denote the probability of a customer purchasing a color television set as p1 = 0.2, the probability of a customer purchasing an ordinary television set as p2 = 0.5, and the probability of a customer just browsing as p3 = 0.3. Then, the probability of two customers purchasing color sets, one customer purchasing an ordinary set, and two customers purchasing nothing can be calculated as follows:

P(X1 = 2, X2 = 1, X3 = 2) = (5 choose 2,1,2) * 0.2^2 * 0.5^1 * 0.3^2 ≈ 0.12075

Here, X1, X2, and X3 are multinomial random variables that represent the number of customers purchasing color sets, ordinary sets, and browsing, respectively. The expression "(5 choose 2,1,2)" represents the number of ways to choose 2 customers who purchase color sets, 1 customer who purchases an ordinary set, and 2 customers who purchase nothing from a total of 5 customers, and can be calculated as follows:

(5 choose 2,1,2) = 5! / (2! * 1! * 2!)

Therefore, the probability of two customers purchasing color sets, one customer purchasing an ordinary set, and two customers purchasing nothing is approximately 0.12075 or 12.08%.

We analyzed the probability of customer purchases based on a given scenario at a television store. Using the multinomial distribution, we calculated the probability of two customers purchasing color sets, one customer purchasing an ordinary set, and two customers purchasing nothing as approximately 12.08%. This suggests that the television store owner should keep a balanced inventory of color and ordinary television sets, and consider offering incentives to browsing customers to increase the chances of a sale.

However, it is important to note that the above calculation assumes that the customers entering the store are independent of each other and have no influence on each other's purchasing decisions, which may not always be the case in practice. Additionally, there may be other factors that affect customer purchases, such as the price, brand, and features of the television sets.