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Chapter 3: Problem 28

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, given that the player is a great hitter.

Short Answer

Expert verified
The probability is denoted as \( P(I|H) \).

Step by step solution

01

Understand the Problem

We are asked to find the probability of a player being an infielder, given that they are a great hitter. This is a conditional probability question, which requires us to use the concept of conditional probability in probability theory.
02

Identify the Events

The events are defined as follows: I is the event that a player is an infielder and H is the event that a player is a great hitter. We want to find the conditional probability of event I given event H.
03

Apply the Conditional Probability Formula

The conditional probability of event I given event H is calculated using the formula: \( P(I|H) = \frac{P(I \cap H)}{P(H)} \), where \( P(I|H) \) is the probability that a player is an infielder given they are a great hitter.
04

Write the Answer using Symbols

Using the symbols identified, the conditional probability we are aiming to find and expresses can be written as \( P(I|H) \). This represents the probability of a player being an infielder, given they are a great hitter.

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Key Concepts

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

Probability Theory
Probability theory is a branch of mathematics that deals with the study of randomness and uncertainty. It provides a framework to model and analyze situations where outcomes are uncertain. In essence, probability theory allows us to predict the likelihood of various events occurring.
A few key elements of probability theory include:
  • **Probability Space**: This is a mathematical construct that models a random experiment. It consists of a sample space, events within this space, and probabilities associated with these events.
  • **Sample Space**: This is the set of all possible outcomes of a random experiment. For example, in a dice roll, the sample space is \( \{1, 2, 3, 4, 5, 6\} \).
  • **Event**: An event is any subset of the sample space. It can be something specific like rolling a 3, or more general like rolling an even number.
  • **Probability of an Event**: This measures how likely it is for the event to occur. It's a number between 0 and 1, where 0 means the event will not occur, and 1 means it definitely will.
This foundational understanding aids in solving problems involving randomness and helps interpret the likelihood of various outcomes in real-world situations.
Event Definition
In probability theory, an event is a specific outcome or a set of outcomes from a random experiment. Events can be simple, like the outcome "heads" in a coin toss, or compound, like the outcome of getting an even number in a dice roll.
Defining an event accurately is crucial to calculating probabilities:
  • **Simple Event**: Represents a single outcome. For example, flipping a coin and getting "heads".
  • **Compound Event**: Consists of two or more simple events. For example, rolling a dice and getting either a 2, 4, or 6.
  • **Mutually Exclusive Events**: These are events that cannot happen simultaneously. For example, rolling a 1 and rolling a 2 on the same die roll.
  • **Independent Events**: The occurrence of one event does not affect the probability of the other. For example, flipping a coin and rolling a die.
Defining events clearly is essential in solving probability problems, as it influences how probabilities are calculated and interpreted.
Bayes' Theorem
Bayes' theorem is one of the fundamental results in probability theory. It provides a way to update our knowledge about the probability of an event based on new evidence. This helps in making informed predictions in uncertain situations.
The theorem is expressed mathematically as:\[P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\] Where:
  • \( P(A|B) \) is the conditional probability of event A occurring given that B is true.
  • \( P(B|A) \) is the probability of event B occurring given that A is true.
  • \( P(A) \) and \( P(B) \) are the probabilities of observing A and B independently.
Bayes' theorem is highly valuable in various fields such as medicine, finance, and machine learning, where it helps in updating predictions or models as more data becomes available.

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Most popular questions from this chapter

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. Complete the table using the data provided. Suppose that one person from the study is randomly selected. Find the probability that person smoked 11 to 20 cigarettes per day.

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.

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.

Consider the following scenario: Let P(C) = 0.4. Let P(D) = 0.5. Let P(C|D) = 0.6. a. Find P(C AND D). b. Are C and D mutually exclusive? Why or why not? c. Are C and D independent events? Why or why not? d. Find P(C OR D). e. Find P(D|C)

In a particular college class, there are male and female students. Some students have long hair and some students have short hair. Write the symbols for the probabilities of the events for parts a through j. (Note that you cannot find numerical answers here. You were not given enough information to find any probability values yet; concentrate on understanding the symbols.) 鈥� Let F be the event that a student is female. 鈥� Let M be the event that a student is male. 鈥� Let S be the event that a student has short hair. 鈥� Let L be the event that a student has long hair. a. The probability that a student does not have long hair. b. The probability that a student is male or has short hair. c. The probability that a student is a female and has long hair. d. The probability that a student is male, given that the student has long hair. e. The probability that a student has long hair, given that the student is male. f. Of all the female students, the probability that a student has short hair. g. Of all students with long hair, the probability that a student is female. h. The probability that a student is female or has long hair. i. The probability that a randomly selected student is a male student with short hair. j. The probability that a student is female.
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