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Chapter 3: Problem 13

A sample space is \(S=\\{a, b, c, d, e\\} .\) Identify two events as \(U=\\{a, b, d\\}\) and \(V=\\{b, c, d\\}\). Suppose \(P(a)\) and \(P(b)\) are each 0.2 and \(P(c)\) and \(P(d)\) are each \(0.1 .\) a. Determine what \(P(e)\) must be. b. Find \(P(U)\). c. Find \(P(V)\)

Short Answer

Expert verified
\(P(e) = 0.4\), \(P(U) = 0.5\), \(P(V) = 0.4\).

Step by step solution

01

Understand the total probability

First, recognize that the sum of the probabilities of all elements in the sample space must equal 1, as it encompasses all possible outcomes. We know that \(S = \{a, b, c, d, e\}\).
02

Calculate the missing probability

Given that \(P(a) = 0.2\), \(P(b) = 0.2\), \(P(c) = 0.1\), and \(P(d) = 0.1\), the total probability of these events is \(0.2 + 0.2 + 0.1 + 0.1 = 0.6\). Therefore, \(P(e)\) must be \(1 - 0.6 = 0.4\).
03

Calculate the probability of event U

Event \(U = \{a, b, d\}\). So, the probability of event \(U\) is calculated by summing the probabilities of the outcomes in \(U\): \(P(U) = P(a) + P(b) + P(d) = 0.2 + 0.2 + 0.1 = 0.5\).
04

Calculate the probability of event V

Event \(V = \{b, c, d\}\). The probability of event \(V\) is calculated by summing the probabilities of the outcomes in \(V\): \(P(V) = P(b) + P(c) + P(d) = 0.2 + 0.1 + 0.1 = 0.4\).

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Key Concepts

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

Sample Space
In probability theory, a sample space is the set of all possible outcomes of an experiment. It is essential to grasp that each possible outcome is distinct and every potential event is accounted for. In our example, we have a sample space denoted as \(S = \{a, b, c, d, e\}\). Each letter represents an individual event or outcome that can occur.
  • "a" through "e" are simply placeholders for different possible results.
  • The complete set \(S\) includes every outcome without omission.
Understanding the sample space is the foundation of analyzing probability because it helps us define probabilities for each specific event. It specifies the universe within which we can compute probabilities for different subsets or events.
Probability of Events
In probability, an event is a subset of the sample space. An event can consist of one or multiple outcomes. Here, events \(U\) and \(V\) are subsets of our sample space \(S\).
  • Event \(U\) includes the outcomes \(\{a, b, d\}\).
  • Event \(V\) includes the outcomes \(\{b, c, d\}\).
To find the probability of these events, you sum the probabilities of the individual outcomes contained in them. This is simply adding together the probability of each outcome that makes up the event. For example:
- The probability of event \(U\) is calculated as \(P(U) = P(a) + P(b) + P(d)\).
- For event \(V\), \(P(V) = P(b) + P(c) + P(d)\).
Probabilities in any sample space will add up to 1, meaning they cover all possible outcomes. This ensures that no event is left unconsidered.
Step-by-Step Solution
A step-by-step solution in probability exercises assists in systematically solving problems using defined methods. Here's how each problem can be approached in a step-by-step manner:
  • Step 1: Begin by understanding the constraints of total probability. All probabilities should sum up to 1 in any sample space.
  • Step 2: Calculate any missing probabilities. Given probabilities for outcomes in \(S = \{a, b, c, d, e\}\), calculate the probability of the missing outcome \(e\) by noting \(P(e) = 1 - (P(a) + P(b) + P(c) + P(d))\).
  • Step 3: Use addition to find the probability of any event, like \(U\) or \(V\), by adding up the probabilities of the outcomes within these events, such as \(P(U)\) or \(P(V)\).
This structured approach helps in breaking down complex problems into manageable, logical steps. It reinforces the understanding of how probabilities interact and depend on the predefined sample space and events configuration.

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Most popular questions from this chapter

Make a statement in ordinary English that describes the complement of each event (do not simply insert the word "not"). a. In the roll of a die: "five or more." b. In a roll of a die: "an even number." c. In two tosses of a coin: "at least one heads." d. In the random selection of a college student: "Not a freshman."

Make a statement in ordinary English that describes the complement of each event (do not simply insert the word "not"). a. In the roll of a die: "two or less." b. In the roll of a die: "one, three, or four." c. In two tosses of a coin: "at most one heads." d. In the random selection of a college student: "Neither a freshman nor a senior."

An accountant has observed that \(5 \%\) of all copies of a particular two-part form have an error in Part I, and \(2 \%\) have an error in Part II. If the errors occur independently, find the probability that a randomly selected form will be error-free.

The sample space that describes the two-way classification of citizens according to gender and opinion on a political issue is $$ S=\\{m f, m a, m n, f f, f a, f n\\}, $$ where the first letter denotes gender \((m:\) male, \(f:\) female \()\) and the second opinion \((f:\) for, \(a\) : against, \(n:\) neutral). For each of the following events in the experiment of selecting a citizen at random, state the complement of the event in the simplest possible terms, then find the outcomes that comprise the event and its complement. a. The person is male. b. The person is not in favor. c. The person is either male or in favor. d. The person is female and neutral.

A tourist who speaks English and German but no other language visits a region of Slovenia. If \(35 \%\) of the residents speak English, \(15 \%\) speak German, and \(3 \%\) speak both English and German, what is the probability that the tourist will be able to talk with a randomly encountered resident of the region?
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