Question

Suppose 5% of Americans have health insurance with the Affordable Care Act (ACA), 85% have other insurance, and 10% have no insurance. Of those covered by the ACA, 55% have a pre-existing condition (such as cancer, diabetes, etc.) 10% of those with other insurance, and 90% of those with no insurance, also have a pre-existing condition.

 (a) (3) Find the probability that a randomly selected person has a pre-existing condition.

 (b) (2) Find the probability that a randomly selected person is covered by the ACA or has a pre-existing condition.

 (c) (2) If everyone who is covered by the ACA and has a pre-existing condition loses their insurance, what will be the proportion of Americans that have no insurance?

 (d) (3) At that point (once (c) has occurred), what will be the probability that a random person has no insurance, given that they have a pre-existing condition?

Answer 1

a)

P(Pre existing condition)=P(ACA)*P(Pre existing condition|ACA)+P(Other)*P(Pre existing condition|Other)+P(No)*P(Pre existing condition|No)

=0.05*0.55+0.85*0.1+0.1*0.9

=0.2025

b)

P(ACA or pre existinig condition)

=P(ACA)+P(pre xisting condition)-P(ACA)*P(Pre existing condition|ACA)

=0.05+0.2025-0.05*0.55

=0.225

c)

new proportion of no insurance =P(previous no insurance proportion)+P(ACA)*P(Pre existing condition|ACA)=0.1+0.05*0.55 =0.1275

d)

P(no insurance | Pre existing condition) =(0.05*0.55+0.1*0.9)/0.2025 =0.5802