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Chapter 3: Problem 38

Use the following information to answer the next two exercises. You are rolling a fair, six-sided number cube. Let E = the event that it lands on an even number. Let M = the event that it lands on a multiple of three. What does P(E|M) mean in words?

Short Answer

Expert verified
The probability of landing on an even number given that it is a multiple of three.

Step by step solution

01

Understand Events E and M

We have two events: event E, which is landing on an even number, and event M, which is landing on a multiple of three when rolling a six-sided die. These events help us identify specific outcomes when rolling the die.
02

Define Conditional Probability

Conditional probability, denoted as \( P(E|M) \), stands for the probability of event E occurring given that event M has occurred. It evaluates how likely it is to roll an even number knowing that the number is a multiple of three.
03

Explain \(P(E|M)\) in Words

Given the context of this problem, \( P(E|M) \) represents the probability of the die landing on an even number if we know it lands on a multiple of three. It expresses the likelihood of event E happening under the condition that event M is already true.

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Key Concepts

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

Understanding Even Numbers on a Six-Sided Cube
When rolling a six-sided number cube, which is essentially a regular die, each side displays a number from 1 to 6. An even number is a number that can be evenly divided by 2 without a remainder. These are important because they help narrow down our outcomes during probability exercises.
- Even numbers on a standard die include: 2, 4, and 6.
- These numbers are crucial to defining event E, which we consider when talking about probabilities related to even numbers.
The purpose of identifying even numbers is to figure out the subset of possible results when the condition (like another event) limits the overall outcomes. Understanding these groupings simplifies solving conditional probability questions.
Identifying Multiples of Three
Multiples of three are numbers that can be expressed as 3 multiplied by another integer. When dealing with a six-sided number cube, recognizing these multiples is key in defining event M in conditional probability problems.
- On a cube with numbers from 1 to 6, the multiples of three are: 3 and 6.
- Event M is distinctly defined by these numbers, providing the possible outcomes where the die shows a multiple of three.
Why is this important? Knowing the multiples helps in calculating probabilities where conditions specify those multiples. By connecting these outcomes with other possible events, like landing on an even number, we create more specific probability scenarios.
The Role of a Six-Sided Number Cube in Probability
A six-sided number cube is the foundation for many basic probability exercises. This cube ensures that there's an equal chance for any number from 1 to 6 to show up. This fairness is essential when calculating probabilities because each number has an equal probability of \(\frac{1}{6}\).
When we deal with events such as landing on an even number or a multiple of three, the cube allows us to define these event probabilities in a seemingly random manner, yet still follow rules of probability. This is possible because each number is distinctly identifiable, letting us analyze the likelihood of any event occurring.
  • For example, a six-sided cube aligns perfectly with the study of conditional probability.
  • It allows us to explore more complex questions beyond the basic outcomes, such as calculating the probability of an event happening given that another event has already taken place.
Therefore, understanding how a six-sided number cube operates within these exercises sheds light on the nuances involved in conditional probability.

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Most popular questions from this chapter

Use the following information to answer the next four exercises. A box is filled with several party favors. It contains 12 hats, 15 noisemakers, ten finger traps, and five bags of confetti. Let H = the event of getting a hat. Let N = the event of getting a noisemaker. Let F = the event of getting a finger trap. Let C = the event of getting a bag of confetti. Find P(C).

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 outfielder and is 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)

Roll two fair dice. Each die has six faces. a. List the sample space. b. Let A be the event that either a three or four is rolled first, followed by an even number. Find P(A). c. Let B be the event that the sum of the two rolls is at most seven. Find P(B). d. In words, explain what 鈥淧(A|B)鈥� represents. Find P(A|B). e. Are A and B mutually exclusive events? Explain your answer in one to three complete sentences, including numerical justification. f. Are A and B independent events? Explain your answer in one to three complete sentences, including numerical justification.

G and H are mutually exclusive events. P(G) = 0.5 P(H) = 0.3 a. Explain why the following statement MUST be false: P(H|G) = 0.4. b. Find P(H OR G). c. Are G and H independent or dependent events? Explain in a complete sentence.
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