45. If a person is over the age of 80, there is a 20% chance of having high blood pressure and a 15% chance of having hearing loss. What is the probability a person over the age of 80 has both diagnoses, assuming that the diagnoses are independent of each other?
Answer: C
The probability a person over the age of 80 has both high blood pressure and hearing loss is 3%.
To find the probability of both diagnoses occurring in a person over the age of 80, we multiply the individual probabilities of each condition, since they are independent. Thus, the calculation is 0.20 (for high blood pressure) multiplied by 0.15 (for hearing loss), which results in 0.03 or 3%.
A) 15%
This option represents the probability of having hearing loss, which is given as 15%. However, it does not account for the joint probability of both conditions occurring simultaneously, thus making it an incorrect choice.
B) 35%
This option incorrectly combines the probabilities of high blood pressure and hearing loss by adding them together. Since the conditions are independent, the correct method is to multiply the probabilities, making this choice invalid.
C) 3%
This option correctly represents the probability of a person over the age of 80 having both high blood pressure and hearing loss. By multiplying the independent probabilities (0.20 * 0.15), the result is indeed 0.03 or 3%.
D) 5%
This option is incorrect as it does not accurately reflect the calculation for the joint probability of both conditions. It may stem from a miscalculation or misunderstanding of how to combine probabilities of independent events.
Conclusion
The correct answer is 3%, as it accurately reflects the probability derived from multiplying the individual probabilities of high blood pressure and hearing loss. All other options miscalculate the relationship between the two independent conditions, either through addition or incorrect multiplication.