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Chapter 4: Problem 45

In an article about investment alternatives, Money magazine reported that drug stocks provide a potential for long-term growth, with over \(50 \%\) of the adult population of the United States taking prescription drugs on a regular basis. For adults age 65 and older, \(82 \%\) take prescription drugs regularly. For adults age 18 to \(64,49 \%\) take prescription drugs regularly. The \(18-64\) age group accounts for \(83.5 \%\) of the adult population (Statistical Abstract of the United States, 2008 ). a. What is the probability that a randomly selected adult is 65 or older? b. Given that an adult takes prescription drugs regularly, what is the probability that the adult is 65 or older?

Short Answer

Expert verified
a. 0.165 b. 0.2485

Step by step solution

01

Define the Problem

We need to find probabilities related to adults taking prescription drugs based on their age groups. This involves finding (a) the proportion of adults aged 65 and older, and (b) the conditional probability that an adult is 65 or older given they take prescription drugs regularly.
02

Calculate Proportion for Part (a)

The entire adult population is divided into two groups: ages 18-64, which is 83.5% of the population, and 65 or older. Thus, the probability that a randomly selected adult is 65 or older is given by the complement: \[ P(65+ \text{ years}) = 1 - 0.835 = 0.165 \].
03

Define Probabilities for Part (b)

Let \( A \) be the event that an adult is 65 or older, \( B \) the event that an adult is 18 to 64, and \( D \) the event that an adult takes prescription drugs regularly. We have:- \( P(A) = 0.165 \)- \( P(B) = 0.835 \)- \( P(D|A) = 0.82 \)- \( P(D|B) = 0.49 \)
04

Use Total Probability Theorem for Overall Probability

Calculate the total probability that an adult takes prescription drugs regularly, \( P(D) \), using the law of total probability:\[P(D) = P(D|A)P(A) + P(D|B)P(B) = 0.82 \times 0.165 + 0.49 \times 0.835 = 0.1353 + 0.40915 = 0.54445\]
05

Calculate Conditional Probability for Part (b)

Use the definition of conditional probability to find \( P(A|D) \):\[ P(A|D) = \frac{P(D|A)P(A)}{P(D)} = \frac{0.82 \times 0.165}{0.54445} \approx 0.2485.\]

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Key Concepts

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

Conditional Probability
Conditional probability is the likelihood of an event occurring given that another event has already occurred. In our exercise, we want to find the probability that an adult is 65 or older, given that they take prescription drugs regularly. This is a conditional probability problem because we need to calculate the chance of an adult's age being 65 and above while knowing they are part of the group that uses prescription drugs.

To calculate conditional probability, we use the formula:
  • \( P(A|B) = \frac{P(A \cap B)}{P(B)} \)
Here, \( A \) represents the condition of being 65 or older, and \( B \) is the condition of taking prescription drugs regularly. In our exercise's solution, we calculated \( P(A|D) \) where \( D \) stands for the drug usage, using the result from the Law of Total Probability (explained next), and the probability of prescription drug usage given the age is 65 or older.
Law of Total Probability
The Law of Total Probability helps us find the probability of an event by considering all possible ways that event can occur. In the context of our problem, we used it to determine the overall probability that an adult takes prescription drugs regularly, incorporating both age groups: 18 to 64, and 65 and older.

This law works especially well when outcomes are partitioned into mutually exclusive events. In our case, these are the two age groups. The formula is:
  • \( P(D) = P(D|A)P(A) + P(D|B)P(B) \)
This computation considers:
  • \( P(D|A) \): Probability that those 65 or older take drugs
  • \( P(A) \): Probability of being 65 or older
  • \( P(D|B) \): Probability that those aged 18 to 64 take drugs
  • \( P(B) \): Probability of being 18 to 64
By adding these weighted probabilities, we find the total probability that someone takes prescription drugs.
Complementary Events
Complementary events are two outcomes of an event that cover all possibilities. For instance, if considering the age group of our population, being 65 or older and being between 18 and 64 are complementary events. This is because if you aren鈥檛 in one group, you must necessarily belong to the other in the context of this problem.

To find the probability of one complementary event, you can subtract the probability of the other event from 1. This is given by:
  • \( P(A') = 1 - P(A) \)
In our exercise, we calculated the probability of someone being 65 or older by using its complementary, which is the probability of being between 18 and 64. Hence, this principle greatly simplifies calculations and provides a clear understanding of how probability distributes among mutually exclusive events.

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Most popular questions from this chapter

The Powerball lottery is played twice each week in 28 states, the Virgin Islands, and the District of Columbia. To play Powerball a participant must purchase a ticket and then select five numbers from the digits 1 through 55 and a Powerball number from the digits 1 through 42\. To determine the winning numbers for each game, lottery officials draw 5 white balls out of a drum with 55 white balls, and 1 red ball out of a drum with 42 red balls. To win the jackpot, a participant's numbers must match the numbers on the 5 white balls in any order and the number on the red Powerball. Eight coworkers at the ConAgra Foods plant in Lincoln, Nebraska, claimed the record \(\$ 365\) million jackpot on February \(18,2006,\) by matching the numbers \(15-17-43-44-49\) and the Powerball number \(29 .\) A variety of other cash prizes are awarded each time the game is played. For instance, a prize of \(\$ 200,000\) is paid if the participant's five numbers match the numbers on the 5 white balls (Powerball website, March 19,2006 ). a. Compute the number of ways the first five numbers can be selected. b. What is the probability of winning a prize of \(\$ 200,000\) by matching the numbers on the 5 white balls? c. What is the probability of winning the Powerball jackpot?

A financial manager made two new investments-one in the oil industry and one in municipal bonds. After a one-year period, each of the investments will be classified as either successful or unsuccessful. Consider the making of the two investments as an experiment. a. How many sample points exist for this experiment? b. Show a tree diagram and list the sample points. c. Let \(O=\) the event that the oil industry investment is successful and \(M=\) the event that the municipal bond investment is successful. List the sample points in \(O\) and in \(M\). d. List the sample points in the union of the events \((O \cup M)\). e. List the sample points in the intersection of the events \((O \cap M)\). f. Are events \(O\) and \(M\) mutually exclusive? Explain.

The prior probabilities for events \(A_{1}, A_{2},\) and \(A_{3}\) are \(P\left(A_{1}\right)=.20, P\left(A_{2}\right)=.50,\) and \(P\left(A_{3}\right)=\) \(.30 .\) The conditional probabilities of event \(B\) given \(A_{1}, A_{2},\) and \(A_{3}\) are \(P\left(B | A_{1}\right)=.50\) \(P\left(B | A_{2}\right)=.40,\) and \(P\left(B | A_{3}\right)=.30\). a. Compute \(P\left(B \cap A_{1}\right), P\left(B \cap A_{2}\right),\) and \(P\left(B \cap A_{3}\right)\). b. Apply Bayes' theorem, equation \((4,19),\) to compute the posterior probability \(P\left(A_{2} | B\right)\). c. Use the tabular approach to applying Bayes' theorem to compute \(P\left(A_{1} | B\right), P\left(A_{2} | B\right)\), and \(P\left(A_{3} | B\right)\) .

Assume that we have two events, \(A\) and \(B\), that are mutually exclusive. Assume further that we know \(P(A)=.30\) and \(P(B)=.40\). a. What is \(P(A \cap B) ?\) b. What is \(P(A | B) ?\) c. \(\quad\) A student in statistics argues that the concepts of mutually exclusive events and independent events are really the same, and that if events are mutually exclusive they must be independent. Do you agree with this statement? Use the probability information in this problem to justify your answer. d. What general conclusion would you make about mutually exclusive and independent events given the results of this problem?

The American Council of Education reported that \(47 \%\) of college freshmen earn a degree and graduate within five years. Assume that graduation records show women make up \(50 \%\) of the students who graduated within five years, but only \(45 \%\) of the students who did not graduate within five years. The students who had not graduated within five years either dropped out or were still working on their degrees. a. \(\quad\) Let \(A_{1}=\) the student graduated within five years \(A_{2}=\) the student did not graduate within five years \(W=\) the student is a female student Using the given information, what are the values for \(P\left(A_{1}\right), P\left(A_{2}\right), P\left(W | A_{1}\right),\) and \\[ P\left(W | A_{2}\right) ? \\] b. What is the probability that a female student will graduate within five years? c. What is the probability that a male student will graduate within five years? d. Given the preceding results, what are the percentage of women and the percentage of men in the entering freshman class?
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