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A First Course in Probability

Sheldon M. Ross

$Math Studyset Vaia Explanations$ Math

9th Edition · 432 pages

ISBN: 9780321794772

Chapter 2: Q. 2.1 (page 52)

Prove the following relations:
$E F \subset E \subset E \cup F$

Short Answer

Expert verified

Use the definition of a subset and logical operations $E F \subset E \subset E \cup F$ .

Step by step solution

01

Given Information.

$E F \subset E \subset E \cup F$

02

Explanation.

$E F \subseteq E \subseteq E \cup F$

If $E F = \emptyset$ , then $E F = \emptyset$ is a subset of all sets by convention.

If $x \in E F$

$By the definition of intersection x \in E and x \in F \to x \in E$

$x$ is a member of $E$ and by the definition of a subset

$A \subseteq B \overset{def}{\Leftrightarrow} x \in A \to x \in B$

$E F \subseteq E$ The same procedure for $E \subset E \cup F$ If $E = \emptyset$ then $E$ is a subset of all sets by convention.

If $x \in E$

$\to x \in E or x \in F$

By the definition of a union $x \in E \cup F$

By the definitionlocalid="1649042050551" $(1)$ , $E \subset E \cup F$

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Most popular questions from this chapter

$\cup_{1}^{\infty} E_{i} F = (\cup_{1}^{\infty} E_{i}) F$ and $\cap_{1}^{\infty} (E_{i} \cup F) = (\cap_{1}^{\infty} E_{i}) \cup F$

For any sequence of events $E 1, E 2, . . .,$ define a new sequence $F 1, F 2, . . .$ of disjoint events (that is, events such that $F i F j = Ø$ whenever $i \neq j$ ) such that for all $n \geq 1$ , $\cup_{1}^{n} F_{i} = \cup_{1}^{n} E_{i}$

Consider the following technique for shuffling a deck of n cards: For any initial ordering of the cards, go through the deck one card at a time, and at each card, flip a fair coin. If the coin comes up heads, then leave the card where it is; if the coin comes up tails, then move that card to the end of the deck. After the coin has been flipped n times, say that one round has been completed. For instance, if $n = 4$ the initial ordering is $1, 2, 3, 4,$ then if the successive flips result in the outcome $h, t, t, h,$ then the ordering at the end of the round is $1, 4, 2, 3 .$ Assuming that all possible outcomes of the sequence of $n$ coin flips are equally likely, what is the probability that the ordering after one round is the same as the initial ordering?

1. A cafeteria offers a three-course meal consisting of an entree, a starch, and a dessert. The possible choices are given in the following table:

Course	Choices
Entree	Chicken or roast beef
Starch	Pasta or rice or potatoes
Dessert	Ice cream or Jello or apple pie or a peach

A person is to choose one course from each category.

$(a)$ How many outcomes are in the sample space?

$(b)$ Let $A$ be the event that ice cream is chosen. How many outcomes are in $A ?$

$(c)$ Let $B$ be the event that chicken is chosen. How many outcomes are in $B ?$

$(d)$ List all the outcomes in the event $A B .$

$(e)$ Let $C$ be the event that rice is chosen. How many outcomes are in $C ?$

$(f)$ List all the outcomes in the event $A B C .$

Prove the following relations:

If $E \subset F,$ then $F^{c} \subset E^{c}$

See all solutions

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