Question

A company has five employees on its health insurance plan. Each year, each employee independently has an 80% probability of no hospital admissions. If an employee requires one or more hospital admissions, the number of admissions is modeled by a geometric distribution with a mean of 1.50. The numbers of hospital admissions of different employees are mutually independent. Each hospital admission costs 20,000.
Calculate the probability that the company's total hospital costs in a year are less than 50,000.

Answer 1

Answer:

the probability that the company's total hospital costs in a year are less than 50,000  = 0.7828

Step-by-step explanation:

From the given information:

the probability that the company's total hospital costs in a year are less than 50,000 will be the sum of the probability of the employees admitted.

If anyone is admitted to the hospital, they have $\dfrac{1}{3}$ probability of making at least one more visit, and a $\dfrac{2}{3}$  probability  that this is their last visit.

If zero employee was admitted ;

Then:

Probability = (0.80)⁵

Probability = 0.3277

If one employee is admitted once;

Probability = $(0.80)^4 \times (0.20)^1 \times (^5_1) \times (\dfrac{2}{3})$

Probability = $(0.80)^4 \times (0.20)^1 \times (\dfrac{5!}{(5-1)!}) \times (\dfrac{2}{3})$

Probability = 0.2731

If one employee is admitted twice

Probability = $(0.80)^3 \times (0.20)^2 \times (^5_2) \times (\dfrac{2}{3})^2$

Probability = $(0.80)^3 \times (0.20)^2 \times (\dfrac{5!}{(5-2)!}) \times (\dfrac{2}{3})^2$

Probability = 0.1820

If two employees are admitted once

Probability = $(0.80)^4\times (0.20)^1 \times (^5_1) \times (\dfrac{1}{3}) \times (\dfrac{2}{3})$

Probability = $(0.80)^4 \times (0.20)^1 \times (\dfrac{5!}{(5-1)!}) \times (\dfrac{1}{3}) \times (\dfrac{2}{3})$

Probability = 0.0910

∴

the probability that the company's total hospital costs in a year are less than 50,000  = 0.3277 + 0.2731 + 0.1820

the probability that the company's total hospital costs in a year are less than 50,000  = 0.7828