91Ó°ÊÓ

In an article about investment growth, Money magazine reported that drug stocks show powerful long-term trends and offer investors unparalleled potential for strong and steady gains. The federal Health Care Financing Administration supports this conclusion through its forecast that annual prescription drug expenditures will reach \(\$ 366\) billion by 2010 , up from \(\$ 117\) billion in 2000 . Many individuals age 65 and older rely heavily on prescription drugs. For this group, \(82 \%\) take prescription drugs regularly, \(55 \%\) take three or more prescriptions regularly, and \(40 \%\) currently use five or more prescriptions. In contrast, \(49 \%\) of people under age 65 take prescriptions regularly, with \(37 \%\) taking three or more prescriptions regularly and \(28 \%\) using five or more prescriptions (Money, September 2001 ). The U.S. Census Bureau reports that of the 281,421,906 people in the United States, 34,991,753 are age 65 years and older (U.S. Census Bureau, Census 2000 ). a. Compute the probability that a person in the United States is age 65 or older. b. Compute the probability that a person takes prescription drugs regularly. c. Compute the probability that a person is age 65 or older and takes five or more prescriptions. d. Given a person uses five or more prescriptions, compute the probability that the person is age 65 or older.

Step by step solution

01

Calculate Probability of Age 65 or Older

To find the probability that a randomly selected person in the United States is age 65 or older, divide the number of people aged 65 or older by the total population. The formula is:\[ \text{Probability} = \frac{\text{Number of people age 65 or older}}{\text{Total population}} = \frac{34,991,753}{281,421,906} \approx 0.1243 \]

02

Calculate Probability of Taking Prescription Drugs Regularly

We need to combine the probabilities for the two age groups: those age 65 or older and those under age 65. Calculate it by:\[ P(\text{prescription}) = P(\text{Age 65+}) \times P(\text{prescription | Age 65+}) + P(\text{Age <65}) \times P(\text{prescription | Age <65}) \]Given:- \(P(\text{prescription | Age 65+}) = 0.82\)- \(P(\text{prescription | Age <65}) = 0.49\)Plugging in the numbers:\[ P(\text{prescription}) = 0.1243 \times 0.82 + (1 - 0.1243) \times 0.49 = 0.1016 + 0.4316 = 0.5332 \]

03

Calculate Probability of Age 65 or Older and Taking Five or More Prescriptions

The probability that a person is age 65 or older and takes five or more prescriptions is calculated by multiplying the probability that they are 65 or older by the probability that given this, they take five or more prescriptions. Thus:\[ P(\text{Age 65+ and 5+ prescriptions}) = P(\text{Age 65+}) \times P(5+ \text{ prescriptions | Age 65+}) \]Given:- \(P(5+ \text{ prescriptions | Age 65+}) = 0.40\)Plugging in the numbers:\[ P(\text{Age 65+ and 5+ prescriptions}) = 0.1243 \times 0.40 = 0.0497 \]

04

Calculate Conditional Probability of Age 65 or Older Given 5+ Prescriptions

We want the probability that a person is age 65 or older given that they already take 5 or more prescriptions. This uses Bayes' Theorem:\[ P(\text{Age 65+} | 5+ \text{ prescriptions}) = \frac{P(\text{Age 65+ and 5+ prescriptions})}{P(5+ \text{ prescriptions})} \]We already found: - \(P(\text{Age 65+ and 5+ prescriptions}) = 0.0497\)Now calculate \(P(5+ \text{ prescriptions})\):\[ P(5+ \text{ prescriptions}) = P(\text{Age 65+}) \times P(5+ \text{ prescriptions | Age 65+}) + P(\text{Age <65}) \times P(5+ \text{ prescriptions | Age <65}) = 0.1243 \times 0.40 + (1 - 0.1243) \times 0.28 \]Calculating:\[ P(5+ \text{ prescriptions}) = 0.04972 + 0.24657 = 0.29629 \]Plug it into the formula:\[ P(\text{Age 65+} | 5+ \text{ prescriptions}) = \frac{0.0497}{0.29629} \approx 0.1678 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already happened. It allows us to refine our predictions based on new information. Let's illustrate with a simple example relevant to the exercise.

Imagine you're asked to find the probability that someone is age 65 or older if you already know they take five or more prescriptions. This probability is not the same as just finding someone aged 65 or older or just taking five medications. It's specifically looking at those who match both criteria simultaneously.

To calculate this, we use Bayes' Theorem. The formula for conditional probability is:
- \[ P(A | B) = \frac{P(A \text{ and } B)}{P(B)} \]
Where:
- \( A \) is the event that the person is age 65 or older.
- \( B \) is the event that the person takes five or more prescriptions.

The numerator \( P(A \text{ and } B) \) represents the joint probability of both events happening together, and the denominator \( P(B) \) is the probability of the event \( B \) occurring.

In the exercise, calculating these probabilities involved understanding the specific age and prescription data provided, helping us to pinpoint an accurate conditional probability.

Age Demographics and Their Role in Predictions

When we talk about age demographics, we're essentially looking at how different age groups are distributed in a population. This type of data is pivotal for understanding various societal trends and needs.

In this exercise, we worked with information about people age 65 and older, which is a significant demographic due to their increased reliance on prescription drugs. The probability of a person being age 65 or older was calculated using the population data from the U.S. Census Bureau:

- Total U.S. population: 281,421,906
- Population age 65 or older: 34,991,753

By dividing these two numbers, we found that approximately 12.43% of people are aged 65 or older. This data is crucial because this age group tends to consume more medication due to potential health issues that come with aging.

Such demographic insights are essential for making predictions not only about healthcare needs but also for planning resource allocation and policy development in sectors like health insurance and social services.

Exploring Prescription Drug Usage Patterns

Prescription drug usage patterns are essential for understanding health trends within various populations. In the context of this exercise, patterns in drug usage help illuminate differences between age groups.

For people aged 65 and older, the usage statistics were quite high. For example:
- 82% regularly take prescription drugs.
- 55% take three or more medications.
- 40% take five or more medications.

In contrast, those under age 65 have different usage patterns:
- 49% take prescriptions regularly.
- 37% take three or more prescriptions.
- 28% use five or more.

These statistics are vital for both healthcare providers and policymakers. For healthcare providers, it impacts decisions on drug stocking and patient guidance. For policymakers, it informs regulations around medication production and distribution, insurance coverage, and even budgeting for public health initiatives.

Having a clear understanding of these patterns ensures that both individual health needs and broader public health objectives can be met effectively, making demographic studies like these extremely valuable to societal planning.