Questions: 68. A hospital's financial reimbursement payor mix indicates 35% of the patients have Medicare and 15% have Medicaid. Of those who have Medicare, there is a 10% chance they also have Medicaid. What is the probability that a hospital patient has both Medicare and Medicaid? 69. 14% of people will contract Virus A over the course of their life, and 18% contract Virus B. 10.62% of people will contract both viruses. What is the probability, to the nearest percent, that a person who contracts Virus B will also contract Virus A?

Solution Steps

To find the probability that a hospital patient has both Medicare and Medicaid, we use the following formula:

\[ P(\text{Medicare} \cap \text{Medicaid}) = P(\text{Medicare}) \times P(\text{Medicaid | Medicare}) \]

Given:

• \( P(\text{Medicare}) = 0.35 \)
• \( P(\text{Medicaid | Medicare}) = 0.10 \)

Thus, we calculate:

\[ P(\text{Medicare} \cap \text{Medicaid}) = 0.35 \times 0.10 = 0.034999999999999996 \]

To determine the probability that a person who contracts Virus B will also contract Virus A, we apply the formula for conditional probability:

\[ P(\text{A | B}) = \frac{P(\text{A} \cap \text{B})}{P(\text{B})} \]

Given:

• \( P(\text{A}) = 0.14 \)
• \( P(\text{B}) = 0.18 \)
• \( P(\text{A} \cap \text{B}) = 0.1062 \)

Thus, we calculate:

\[ P(\text{A | B}) = \frac{0.1062}{0.18} = 0.5899999999999999 \]

To express the conditional probability \( P(\text{A | B}) \) as a percentage, we multiply by 100:

\[ P(\text{A | B}) \times 100 = 0.5899999999999999 \times 100 = 59\% \]

Final Answer