9.7 Probability (Page 3/18)

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

$\frac{7}{13}$

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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}

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Try It

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

$\frac{2}{13}$

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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.
$\begin{array}{l} P (E^{'}) = 1 - P (E) \\ = 1 - \frac{1}{12} \\ = \frac{11}{12} \end{array}$

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Questions & Answers

If c is the cost function for a particular product, find the marginal cost functions and their values at x=10 a. c(x) = 800+ 0.04x + 0.0002x² b. c(x) = 250 + 100x + 0.001x²

Mamush Reply

how can I find set theory

Ephraim Reply

how can I find set theory

Jarvis

is there an error on the one about the dime's thickness? says 2.2x10⁶=0.00135 m

Patrick Reply

hi, interested in algebra

Makan Reply

how to reduce an equation?

Makan

by manipulation of both side

9(y+8)-27 is 9y+45. Why can't you reduce that to y+5? I know that's wrong but can't explain why

Patrick Reply

when you reduce an equation to its simplest terms, you can't change the value of the equation. reducing it to y + 5 is equivalent to dividing it by 9 which changes the value. you can multiply it by 1 or 9/9 which would give 9(y + 5). multiplying it by one does not change the value.

Philip

Given a polynomial expression, factor out the greatest common factor.

Hanu Reply

WHAT IS QUADRATIC EQUATION?

Charles Reply

WHAT IS SYSTEM OF LINEAR INEWUALITIES?

Charles

WHAT IS SYSTEM OF LINEAR INEWUALITIES?

Charles

complex perform

Angel

what is equation?

Charles Reply

what are equations?

Charles

Definition of economics according to karl Marx Thomas malthus Jeremy bentham David Ricardo J.K

Rakiya

Please help me is assignment

Rakiya

The 47th problem of Euclid

Kenneth

show that the set of all natural number form semi group under the composition of addition

Nikhil Reply

what is the meaning

Dominic

explain and give four Example hyperbolic function

Lukman Reply

_3_2_1

felecia

⅗ ⅔½

felecia

_½+⅔-¾

felecia

The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu

SABAL Reply

1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3

Pawel

2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31

Pawel

Ifeanyi

on number 2 question How did you got 2x +2

Ifeanyi

combine like terms. x + x + 2 is same as 2x + 2

Pawel

x*x=2

felecia

2+2x=

felecia

×/×+9+6/1

Debbie

Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22

Naagmenkoma

Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?

mariel Reply

Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113

Pawel

how do I set up the problem?

Harshika Reply

what is a solution set?

Harshika

find the subring of gaussian integers?

Rofiqul

hello, I am happy to help!

Shirley Reply

please can go further on polynomials quadratic

Abdullahi

hi mam

Mark

I need quadratic equation link to Alpa Beta

Abdullahi Reply

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

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