9.7 Probability (Page 3/18)

<< Chapter < Page

College algebra

Page 3 / 18

Chapter >> Page >

Try It

A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.

$\frac{7}{13}$

Got questions? Get instant answers now!

Computing the probability of mutually exclusive events

Suppose the spinner in [link] is spun again, but this time we are interested in the probability of spinning an orange or a $d .$ There are no sectors that are both orange and contain a $d,$ so these two events have no outcomes in common. Events are said to be mutually exclusive events when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is

P (E \cup F) = P (E) + P (F)

Notice that with mutually exclusive events, the intersection of $E$ and $F$ is the empty set. The probability of spinning an orange is $\frac{3}{6} = \frac{1}{2}$ and the probability of spinning a $d$ is $\frac{1}{6} .$ We can find the probability of spinning an orange or a $d$ simply by adding the two probabilities.

\begin{array}{l} P (E \cup^{​} F) = P (E) + P (F) \\ = \frac{1}{2} + \frac{1}{6} \\ = \frac{2}{3} \end{array}

The probability of spinning an orange or a $d$ is $\frac{2}{3} .$

A General Note

Probability of the union of mutually exclusive events

The probability of the union of two mutually exclusive events $E and F$ is given by

P (E \cup F) = P (E) + P (F)

How To

Given a set of events, compute the probability of the union of mutually exclusive events.

Determine the total number of outcomes for the first event.
Find the probability of the first event.
Determine the total number of outcomes for the second event.
Find the probability of the second event.
Add the probabilities.

Computing the probability of the union of mutually exclusive events

A card is drawn from a standard deck. Find the probability of drawing a heart or a spade.

The events “drawing a heart” and “drawing a spade” are mutually exclusive because they cannot occur at the same time. The probability of drawing a heart is $\frac{1}{4},$ and the probability of drawing a spade is also $\frac{1}{4},$ so the probability of drawing a heart or a spade is

\frac{1}{4} + \frac{1}{4} = \frac{1}{2}

Got questions? Get instant answers now!

Try It

A card is drawn from a standard deck. Find the probability of drawing an ace or a king.

$\frac{2}{13}$

Got questions? Get instant answers now!

Using the complement rule to compute probabilities

We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will not happen. The complement of an event $E,$ denoted $E^{'},$ is the set of outcomes in the sample space that are not in $E .$ For example, suppose we are interested in the probability that a horse will lose a race. If event $W$ is the horse winning the race, then the complement of event $W$ is the horse losing the race.

To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1.

P (E^{'}) = 1 - P (E)

The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is $\frac{1}{9},$ the probability of the horse losing the race is simply

1 - \frac{1}{9} = \frac{8}{9}

A General Note

The complement rule

The probability that the complement of an event will occur is given by

P (E^{'}) = 1 - P (E)

Using the complement rule to calculate probabilities

Two six-sided number cubes are rolled.

Find the probability that the sum of the numbers rolled is less than or equal to 3.
Find the probability that the sum of the numbers rolled is greater than 3.

The first step is to identify the sample space, which consists of all the possible outcomes. There are two number cubes, and each number cube has six possible outcomes. Using the Multiplication Principle, we find that there are $6 \times 6,$ or $36$ total possible outcomes. So, for example, 1-1 represents a 1 rolled on each number cube.

$1-1$	$1-2$	$1-3$	$1-4$	$1-5$	$1-6$
$2-1$	$2-2$	$2-3$	$2-4$	$2-5$	$2-6$
$3-1$	$3-2$	$3-3$	$3-4$	$3-5$	$3-6$
$4-1$	$4-2$	$4-3$	$4-4$	$4-5$	$4-6$
$5-1$	$5-2$	$5-3$	$5-4$	$5-5$	$5-6$
$6-1$	$6-2$	$6-3$	$6-4$	$6-5$	$6-6$

We need to count the number of ways to roll a sum of 3 or less. These would include the following outcomes: 1-1, 1-2, and 2-1. So there are only three ways to roll a sum of 3 or less. The probability is
$\frac{3}{36} = \frac{1}{12}$
Rather than listing all the possibilities, we can use the Complement Rule. Because we have already found the probability of the complement of this event, we can simply subtract that probability from 1 to find the probability that the sum of the numbers rolled is greater than 3.
$\begin{array}{l} P (E^{'}) = 1 - P (E) \\ = 1 - \frac{1}{12} \\ = \frac{11}{12} \end{array}$

Got questions? Get instant answers now!

Questions & Answers

differentiate between demand and supply giving examples

Lambiv Reply

differentiated between demand and supply using examples

Lambiv

what is labour ?

Lambiv

how will I do?

Venny Reply

how is the graph works?I don't fully understand

Rezat Reply

information

Eliyee

devaluation

Eliyee

WARKISA

hi guys good evening to all

Lambiv

multiple choice question

Aster Reply

appreciation

Eliyee

explain perfect market

Lindiwe Reply

In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

What is ceteris paribus?

Shukri Reply

other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

Kelo

yes,thank you

Shukri

Can I ask you other question?

Shukri

what is monopoly mean?

Habtamu Reply

What is different between quantity demand and demand?

Shukri Reply

Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

Ezea

Shukri

how do you save a country economic situation when it's falling apart

Lilia Reply

what is the difference between economic growth and development

Fiker Reply

Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

Shukri

production function means

Jabir

What do you think is more important to focus on when considering inequality ?

Abdisa Reply

any question about economics?

Awais Reply

sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏

Asui

it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

Awais

thank you so much 👍 sir

Asui

In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

Cornelius

Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

Feyisa Reply

Answer

Feyisa

Jabir

the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema

Gsbwnw Reply

suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product

Abdureman

types of unemployment

Yomi Reply

What is the difference between perfect competition and monopolistic competition?

Mohammed

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 9

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask

	5 Using the complement rule to compute probabilities By OpenStax Read Online Course
	3 Computing the probability of the union of two events By OpenStax Read Online Course
	Corporate Communication BUS210 By P. Wynn Norman Start Quiz
	Clinical Issues in TB Management By Cath Yu Start Quiz
©flickr:	Anatomy Physiology By Jemekia Weeden Start Quiz
	11 Clin Sci I - GI Twedt quiz 2 By Brooke Delaney Start Exam
	3 Physiotherapy Flashcards Set 3 By Rhodes Start Flashcards
	2 Timeshare 2 By Jams Kalo Start Quiz
	3 AP 03 Cellular Level of Organization MCQ By OpenStax Start Quiz
	7 Biology 07 Cellular Respiration MCQ By OpenStax Start Quiz
	28 Biology 28 Invertebrates MCQ By OpenStax Start Quiz
	6 Biology 06 Metabolism MCQ By OpenStax Start Quiz