Question

The probabilities associated with the expected principal source of payment for hospital discharges in the United States in the year 1990 are listed below:

Principle source of Payment Probability

Private Insurance 0.412

Medicare 0.325

Medicaid 0.107

Other Govt. Program ????

Self-payment 0.039

Other/ No charge 0.033

Not stated 0.025

1. What is the probability that the principal source of payment is Other Govt. Program? Which answer is correct? There might be slight rounding differences. Please provide an explanation.

a. 0.001

b. 0.059

c. 0.905

d. 0.509

2. What is the probability that the patient’s principal source of payment is Medicare, Medicaid, or some other government program? Which answer is correct? There might be slight rounding differences. Please provide an explanation.

a. 0.001

b. 0.941

c. 0.149

d. 0.491

3. Given that the principal source of payment is NOT a government program, what is the probability that it is Private Insurance? Which answer is correct? There might be slight rounding differences. Please provide an explanation.

a. 0.321

b. 0.980

c. 0.809

d. 0.123

Answer 1

Solution:

Given:

Principle source of Payment	Probability
Private Insurance	0.412
Medicare	0.325
Medicaid	0.107
Other Govt. Program	????
Self-payment	0.039
Other/ No charge	0.033
Not stated	0.025

Part 1) What is the probability that the principal source of payment is Other Govt. Program?

P( Other Govt. Program)=............?

Total probability should be 1, that is:

0.412+ 0.325 +0.107 +P( Other Govt. Program) +0.039 +0.033 +0.025 = 1.000

0.941 + P( Other Govt. Program) = 1.000

P( Other Govt. Program) = 1.000 - 0.941

P( Other Govt. Program) = 0.059

Thus correct answer is:

b. 0.059

Part 2) What is the probability that the patient’s principal source of payment is Medicare, Medicaid, or some other government program?

P( Medicare, Medicaid, or some other government program) =.............?

P( Medicare, Medicaid, or some other government program) =P( Medicare)+ P(Medicaid)+P(other government program)

P( Medicare, Medicaid, or some other government program) =0.325 + 0.107 + 0.059

P( Medicare, Medicaid, or some other government program) = 0.491

Thus correct answer is:

d. 0.491

Part 3) Given that the principal source of payment is NOT a government program, what is the probability that it is Private Insurance?

P( Private Insurance | NOT a government program ) =.................?

P( Private Insurance | NOT a government program )

=P( Private Insurance and NOT a government program ) / P( NOT a government program )

= P( Private Insurance) / [1 - P( a government program ) ]

= P( Private Insurance) / [ 1 - P(Medicare) + P(Medicaid) + P (other government program) ]

P( Private Insurance | NOT a government program ) = 0.412 / [ 1 - ( 0.325 + 0.107 +0.059) ]

P( Private Insurance | NOT a government program ) = 0.412 / [ 1 -0.491 ]

P( Private Insurance | NOT a government program ) = 0.412 / 0.509

P( Private Insurance | NOT a government program ) = 0.809

Thus correct answer is:

c. 0.809