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Chapter 3: Problem 128

Answer these questions using probability rules. Do NOT use the contingency table. Three thousand fifty-nine cases of AIDS had been reported in Santa Clara County, CA, through a certain date. Those cases will be our population. Of those cases, 6.4% obtained the disease through heterosexual contact and 7.4% are female. Out of the females with the disease, 53.3% got the disease from heterosexual contact. a. Find P(Person is female). b. Find P(Person obtained the disease through heterosexual contact). c. Find P(Person is female GIVEN person got the disease from heterosexual contact) d. Construct a Venn diagram representing this situation. Make one group females and the other group heterosexual contact. Fill in all values as probabilities.

Short Answer

Expert verified
a. 0.074, b. 0.064, c. 0.533

Step by step solution

01

Calculate P(Person is female)

Given that 7.4% of the 3059 reported AIDS cases are female, we calculate the probability that a randomly chosen person is female as follows:\[ P(\text{Female}) = \frac{7.4}{100} = 0.074 \]
02

Calculate P(Person obtained the disease through heterosexual contact)

We know that 6.4% of the cases obtained the disease through heterosexual contact. Thus, the probability is calculated as:\[ P(\text{Heterosexual contact}) = \frac{6.4}{100} = 0.064 \]
03

Calculate P(Person is female GIVEN person got the disease from heterosexual contact)

We are given that out of all females with the disease, 53.3% got the disease from heterosexual contact. We use the condition provided to find the conditional probability:\[ P(\text{Female} \,|\, \text{Heterosexual contact}) = \frac{53.3}{100} = 0.533 \]
04

Construct a Venn Diagram

To construct the Venn diagram, create two intersecting circles, one representing females and the other representing those who obtained the disease through heterosexual contact. Mark the following probabilities: - Total probability of females: 0.074 - Total probability of heterosexual contact: 0.064 - Intersection probability (females and heterosexual contact): 0.533 of the females, which is (0.533 * 0.074) = 0.0394.

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Key Concepts

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

Conditional Probability
Conditional probability allows us to calculate the likelihood of an event occurring, given that another event has already occurred. In simpler terms, it tells us how probable an event is when we have additional information about the circumstances surrounding it.
For example, in the context of the AIDS statistics provided, if we know that an individual obtained the disease through heterosexual contact, we want to determine the probability that this individual is a female given that information. This is different from simply calculating the probability of a person being female overall.
The formula for conditional probability is:
  • If "A" and "B" are two events, then the probability of "A" given "B" has occurred is: \[ P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)} \]
This formula lets us tailor our probability assessment based on known conditions, thereby providing more precise estimates for specific situations.
Venn Diagram
A Venn diagram is a visual tool used to depict relationships between different sets of data. It鈥檚 a simple way to represent probabilities and understand the overlap between different categories or groups.
In the case of the AIDS statistics problem, two key groups were considered: females and those who obtained the disease through heterosexual contact. By drawing two intersecting circles, one can visually represent these groups. The region where the circles overlap shows the probability of a case that falls into both categories simultaneously.
Here's a quick guide to constructing a Venn diagram for this scenario:
  • Total area of each circle: Represents the total probability of each group.
  • Intersection: This section shows the combined probability (or overlap) of belonging to both groups. For this problem, it's the probability that a person is both female and acquired the disease through heterosexual contact.
Venn diagrams help to clearly visualize and more easily analyze probabilities that arise from overlapping categories.
AIDS Statistics
Analyzing real-world data, like AIDS statistics, aids in understanding how probability applies to complex, societal issues. Challenges such as disease transmission require grasping nuanced probability rules. Santa Clara County, with its specific data, provides an excellent example.
From the provided data:
  • Reported cases: 3059 cases represent the total scope of the data set.
  • Heterosexual contact: This mode of transmission accounts for 6.4% of all cases.
  • Female cases: Constituting 7.4% of all reported cases within the county.
  • Females and heterosexual contact: Of the female cases, 53.3% contracted it through heterosexual activity.
Such information supports calculations and interpretations that can aid in public health efforts and understanding disease spread dynamics.
Probability Calculation
Probability calculation is essential in assessing the risk or likelihood of different outcomes. In the AIDS statistics problem, understanding how to compute probabilities assists in making informed decisions.
The given percentages allow for calculating probabilities by
  • Converting percentage values into decimals (e.g., 7.4% becomes 0.074) when calculating probabilities.
  • Considering intersections, where events overlap, using multiplication of probabilities.
For instance, the calculation for the intersection of females who contract AIDS via heterosexual contact uses this principle: multiply the probability of being female by the probability of contracting the disease this way. Thus, you calculate a specific and detailed probability that informs public health policy and personal awareness.

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Most popular questions from this chapter

On February 28, 2013, a Field Poll Survey reported that 61% of California registered voters approved of allowing two people of the same gender to marry and have regular marriage laws apply to them. Among 18 to 39 year olds (California registered voters), the approval rating was 78%. Six in ten California registered voters said that the upcoming Supreme Court鈥檚 ruling about the constitutionality of California鈥檚 Proposition 8 was either very or somewhat important to them. Out of those CA registered voters who support same-sex marriage, 75% say the ruling is important to them. In this problem, let: 鈥� C = California registered voters who support same-sex marriage. 鈥� B = California registered voters who say the Supreme Court鈥檚 ruling about the constitutionality of California鈥檚 Proposition 8 is very or somewhat important to them 鈥� A = California registered voters who are 18 to 39 years old. a. Find P(C). b. Find P(B). c. Find P(C|A). d. Find P(B|C). e. In words, what is C|A? f. In words, what is B|C? g. Find P(C AND B). h. In words, what is C AND B? i. Find P(C OR B). j. Are C and B mutually exclusive events? Show why or why not.

Use the following information to answer the next seven exercises. An article in the New England Journal of Medicine, reported about a study of smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans, and 7,650 Whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 Whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 Whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 Whites. In words, explain what it means to pick one person from the study who is 鈥淛apanese American AND smokes 21 to 30 cigarettes per day.鈥� Also, find the probability.

Use the following information to answer the next ten exercises. On a baseball team, there are infielders and outfielders. Some players are great hitters, and some players are not great hitters. Let I = the event that a player in an infielder. Let O = the event that a player is an outfielder. Let H = the event that a player is a great hitter. Let N = the event that a player is not a great hitter Write the symbols for the probability that a player is an infielder and is not a great hitter.

The following table of data obtained from www.baseball-almanac.com shows hit information for four players. Suppose that one hit from the table is randomly selected. $$\begin{array}{|l|l|l|l|l|}\hline \text { Name } & {\text { single }} & {\text { Double }} & {\text { Triple }} & {\text { Home Run }} & {\text { Total Hits }} \\ \hline \text { Babe Ruth } & {1,517} & {506} & {136} & {714} & {2,873} \\ \hline \text { Jackie Robinson } & {1,054} & {273} & {54} & {137} & {1,518} \\ \hline \text { Ty Cobb } & {3,603} & {174} & {295} & {114} & {4,189} \\ \hline \text { Hank Aaron } & {2,294} & {624} & {98} & {755} & {3,771} \\ \hline\end{array}$$ Are "the hit being made by Hank Aaron" and "the hit being a double" independent events? a. Yes, because P(hit by Hank Aaron|hit is a double) = P(hit by Hank Aaron) b. No, because P(hit by Hank Aaron|hit is a double) ? P(hit is a double) c. No, because P(hit is by Hank Aaron|hit is a double) ? P(hit by Hank Aaron) d. Yes, because P(hit is by Hank Aaron|hit is a double) = P(hit is a double)

Use the following information to answer the next six exercises. A jar of 150 jelly beans contains 22 red jelly beans, 38 yellow, 20 green, 28 purple, 26 blue, and the rest are orange. Let B = the event of getting a blue jelly bean Let G = the event of getting a green jelly bean. Let O = the event of getting an orange jelly bean. Let P = the event of getting a purple jelly bean. Let R = the event of getting a red jelly bean. Let Y = the event of getting a yellow jelly bean. Find P(R).
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