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Chapter 3: Problem 27

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 a great hitter, given that the player is an infielder.

Short Answer

Expert verified
The symbol is \( P(H|I) \).

Step by step solution

01

Identify known probabilities

The question refers to the probability of a player being a great hitter given that they are an infielder. We need to identify the events involved: - Event \( H \) refers to a player being a great hitter.- Event \( I \) refers to a player being an infielder.
02

Recognize conditional probability

The question asks for the probability of event \( H \) (a player being a great hitter) given event \( I \) (a player is an infielder). This kind of probability refers to conditional probability.
03

Express conditional probability symbolically

In probability, the notation for the conditional probability of event \( H \) given event \( I \) is written as \( P(H|I) \). This is read as "the probability of \( H \) given \( I \)."

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Key Concepts

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

Probability Notation
Probability notation is a way to represent different probability concepts using specific symbols. These symbols help to simplify the communication of probability ideas.
Quite often, probability notation can transform complex ideas into something more understandable.In the context of our exercise, different letters are used to represent events:
  • Event \( I \) is the occurrence of a player being an infielder.
  • Event \( O \) symbolizes a player being an outfielder.
  • Event \( H \) corresponds to a player being a great hitter.
  • Event \( N \) refers to a player not being a great hitter.

The conditional probability notation we discussed, \( P(H|I) \) , succinctly represents "the probability that event \( H \) occurs, given that event \( I \) has occurred."
Understanding these notations can make it much easier to navigate through probability exercises and analyses.
Probability Events
A probability event is any outcome or combination of outcomes from a probability experiment.
Events can be anything from tossing a coin to drawing a card from a deck.In the baseball team scenario, events are classified by characteristics of players:
  • **Infielders (I)** are part of one event.
  • **Outfielders (O)** belong to another.
  • **Great hitters (H)** represent a separate event based on batting capabilities.
  • **Not great hitters (N)** form the opposite of event \( H \).
Each event can occur alone or in combination with others, like a player being both an infielder and a great hitter. In probability, understanding how each event behaves and interacts with others is crucial for accurately calculating outcomes.
Conditional Events
Conditional events refer to scenarios where the probability of an event is influenced by the occurrence of another event.
This introduces the concept of conditional probability, which is essential for more advanced probability problems.In our example, we looked at the conditional event of being a great hitter \( (H) \) given that the player is an infielder \( (I) \).
The notation \( P(H|I) \) helps us comprehend how the restriction (being an infielder) affects the probability.Calculating conditional probabilities can involve new approaches and formulas such as:
  • Using the formula \( P(H|I) = \frac{P(H \cap I)}{P(I)} \), where \( P(H \cap I) \) represents the probability of both events occurring together.
Conditional probabilities allow us to capture nuanced probability scenarios, especially when events are dependent on specific criteria or conditions, making them powerful tools in data analysis and predictions.

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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. In words, explain what it means to pick one person from the study who is 鈥淛apanese American OR smokes 21 to 30 cigarettes per day.鈥� Also, find the probability.

An experiment consists of first rolling a die and then tossing a coin. a. List the sample space. b. Let A be the event that either a three or a four is rolled first, followed by landing a head on the coin toss. Find P(A). c. Let B be the event that the first and second tosses land on heads. Are the events A and B mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification.

What is the probability of drawing a club in a standard deck of 52 cards?

Use the following information to answer the next two exercises. The percent of licensed U.S. drivers (from a recent year) that are female is 48.60. Of the females, 5.03% are age 19 and under; 81.36% are age 20鈥�64; 13.61% are age 65 or over. Of the licensed U.S. male drivers, 5.04% are age 19 and under; 81.43% are age 20鈥�64; 13.53% are age 65 or over. Suppose that 10,000 U.S. licensed drivers are randomly selected. a. How many would you expect to be male? b. Using the table or tree diagram, construct a contingency table of gender versus age group. c. Using the contingency table, find the probability that out of the age 20鈥�64 group, a randomly selected driver is female.

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)
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