Why is the sum of probability is not equal to 1 in a geometric distribution?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.The probability of each employee has at least one hospital admissions is ( 0.2 ⋅ 1 ) / ( 1 − 2 / 3 ) = 0.6 instead of 0.2. Why is that?

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Why is the sum of probability is not equal to 1 in a geometric distribution?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.The probability of each employee has at least one hospital admissions is   (  0.2  ⋅  1  )      /    (  1  −  2      /    3  )  =  0.6 instead of 0.2. Why is that?

Joel Reese · Accepted Answer

Step 1If the employee has a hospital admission, the number of admissions is geometrically distributed with mean       3    2  . That is the probability of exactly k admissions is   p  (  1  −  p      )          k      −      1        ,     k  =  1  ,  2  ,  … and the mean is       1    p   so   p  =      2    3    ..Step 2However, these probabilities are conditional probabilities, contingent on the employee&#039;s having had at least one admission (probability .2.) Therefore the probability       p    k   that an employee has exactly k admissions is       p    k    =      {                            .8          ,                                                    k              =              0                                                            .4          ⋅                      3                          −              k                                ,                                                    k              &amp;gt;              0

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