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Chapter 5: Problem 5

Let the sample space be \(S=\\{1,2,3,4,5,6,7,8,9,10\\} .\) Suppose that the outcomes are equally likely. Compute the probability of the event: \(E=\\{1,3,5,10\\}\)

Short Answer

Expert verified
The probability is \(\frac{2}{5}\).

Step by step solution

01

- Understand the Given Sample Space

The sample space provided is \(S = \{1,2,3,4,5,6,7,8,9,10\}\).This represents all possible outcomes.
02

- Identify the Event

The event provided is \(E = \{1,3,5,10\}\).This means we are interested in these specific outcomes within the sample space.
03

- Determine Number of Elements

Count the number of elements in the sample space, \(S\). Thus, \(|S| = 10\).Also, count the elements in the event \(E\), so \(|E| = 4\).
04

- Calculate the Probability

Utilize the probability formula for equally likely outcomes: \[ P(E) = \frac{|E|}{|S|}\].Substitute the counted elements: \[ P(E) = \frac{4}{10} = \frac{2}{5} \]

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Key Concepts

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

Sample Space
In probability, the **sample space** of an experiment or random trial is the set of all possible outcomes or results. For example, in the given problem, the sample space is defined as: \( S = \{1,2,3,4,5,6,7,8,9,10\} \). This means there are 10 possible outcomes when conducting the experiment. Understanding the sample space is crucial as it lays the foundation for calculating probabilities. It helps identify all the possible outcomes and ensures none are overlooked.
Event
An **event** is a specific set of outcomes within the sample space. These outcomes are the ones we are particularly interested in. For instance, in the problem, the event is given by: \( E = \{1,3,5,10\} \). This means we are focused on the outcomes where 1, 3, 5, or 10 occur. An event might consist of a single outcome or multiple outcomes. Understanding what constitutes an event is essential because it helps in determining the probability of its occurrence by comparing it to the sample space.
Equally Likely Outcomes
When we say outcomes are **equally likely**, we mean that each outcome has the same chance of happening. In the context of our problem, every number in the sample space \( S = \{1,2,3,4,5,6,7,8,9,10\} \) is equally likely to occur. This makes our calculation straightforward, as each has an equal probability of \( \frac{1}{10} \). The concept of equally likely outcomes is fundamental in probability because it simplifies the calculation of probabilities by allowing us to divide the number of favorable outcomes by the total number of possible outcomes.
Counting Elements in a Set
To determine probabilities, we often need to count the number of elements in different sets. In our problem, we count the elements in both the sample space and the event. The sample space \( S \) has 10 elements, as shown: \( |S| = 10 \). The event \( E \) has 4 elements: \( |E| = 4 \). This counting allows us to use the probability formula: \( P(E) = \frac{|E|}{|S|} \). By substituting the counts, we get: \( P(E) = \frac{4}{10} = \frac{2}{5} \). Counting elements is a powerful tool in probability as it helps bridge the abstract concepts with concrete numbers.

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Most popular questions from this chapter

According to the Sefton Council Password Policy (August 2007), the United Kingdom government recommends the use of 鈥淓nviron passwords with the following format: consonant, vowel, consonant, consonant, vowel, consonant, number, number (for example, pinray45).鈥� (a) Assuming passwords are not case sensitive, how many such passwords are possible (assume that there are 5 vowels and 21 consonants)? (b) How many passwords are possible if they are case sensitive?

In 1991 , columnist Marilyn Vos Savant posted her reply to a reader's question. The question posed was in reference to one of the games played on the gameshow Let's Make a Deal hosted by Monty Hall. Her reply generated a tremendous amount of backlash, with many highly educated individuals angrily responding that she was clearly mistaken in her reasoning. (a) Using subjective probability, estimate the probability of winning if you switch. (b) Load the Let's Make a Deal applet at www.pearsonhighered.com/sullivanstats. Simulate the probability that you will win if you switch by going through the simulation at least 100 times. How does your simulated result compare to your answer to part (a)? (c) Research the Monty Hall Problem as well as the reply by Marilyn Vos Savant. How does the probability she gives compare to the two estimates you obtained? (d) Write a report detailing why Marilyn was correct. One approach is to use a random variable on a wheel similar to the one shown. On the wheel, the innermost ring indicates the door where the car is located, the middle ring indicates the door you selected, and the outer ring indicates the door(s) that Monty could show you. In the outer ring, green indicates you lose if you switch while purple indicates you win if you switch.

Shawn is planning for college and takes the SAT test. While registering for the test, he is allowed to select three schools to which his scores will be sent at no cost. If there are 12 colleges he is considering, how many different ways could he fill out the score report form?

You are dealt 5 cards from a standard 52-card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering the following questions: (a) How many ways can 5 cards be selected from a 52-card deck? (b) Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck? (c) The remaining 2 cards must be different from the 3 chosen and different from each other. For example, if we drew three kings, the 4th card cannot be a king. After selecting the three of a kind, there are 12 different ranks of card remaining in the deck that can be chosen. If we have three kings, then we can choose twos, threes, and so on. Of the 12 ranks remaining, we choose 2 of them and then select one of the 4 cards in each of the two chosen ranks. How many ways can we select the remaining 2 cards? (d) Use the General Multiplication Rule to compute the probability of obtaining three of a kind. That is, what is the probability of selecting three of a kind and two cards that are not like?

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