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ftFind probabilities of compound eventa.

VocabularyThe rmion or intersection of two events is called a compound event.

Two events are overlapping ifthey have one or more outcomes incommon.

Two events are disjoin! or nutually exclusive, if they have nooutcomes in common.

ffiAEEE Find probability of disioint events

d card is randomly s€lected from a standard deck of 52 Gards. What is

the probability that it is a 5 or an ace?

Let event I be selecting a 5 and event B be selecting an ace. I has 4 outcomes andB has 4 outcomes. Because I and B are disjoint, the probability is:

^ ,L:a=Z-nrsaP(A or B) = P(A) + P(B) : 52 '

52 sz 13

mf|sf| Find probability of overlapping events

A card is randomly selected from a standard deck of 52 cards. l,lfhat isthe probability that it is a club or a 3?

Let event I be selecting a club and event B be selecting a 3. I has 13 outcomes and. -B has 4 outcomes. Of these, I outcome is common to I and B. fiie probability ofselecting a club ora3 is:

P\A or B) : P(A) + P(B) - P(Aand B) : E - + - + : E : fr = o.:os

Use a formula to find PIA and Bl

Given P(Al = o.3, P(8) = O-72' and HA or Et = O.6, find P(A and al'

P(A or B) = P(A) + P(B\ - P(A nd B\ write gbneral formula.

0.6 : 0.3 + 0.72 - P(A and B)

P(A and' B) : 0.42

Exercises for Examples 1, 2, and 3

A card is randomly selected from a standard deck of 52 Gards' Find the

probability of the given event.

1. Selecting a queen or a 4 2. Selecting a spade or a 5

A|gebra 2 3. Find P(,4 and B) when P(l) : 0.25, P(B) : 0.40, and P(A orB) : 0.55.

Chaoter 10 Resource Book

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Substitute known probabilities.

Solve for P(l and B).

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e f- -r'.'i-l PJ"Y#J" F,:i!*"'"'0""0ffi Find probabilities of oomplements

When two six-sided dice are ro11e4 there are 36 possible outcomes' Find the

probability of the given went.

a. The sum is less than or equal to 3. ,

b. The sum is greater than 3.

Solutiona. The outcomes for which the sum is less than or equal to 3 are

(1,r) ,Q,1), and(1,2).

P(sum < 3) : 36: i- 0'083

b. P(sum>3)= I - P(sum(3). t_ r_Tl

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- 0.917

I ffiffi Use a comPlement in real life

: 1 - 0.834

= 0.166

Exercises for ExamPles 4 and 5

5. P( l ) : *

7. P(A):0.0e

8. ln Example 5 if the probability that the recipients of PhD degrees had annuat

salaries in excess of$95,000 was 0.668, what is the probability that a recipient

ftom the study had an annual salary of $95,000 or less?

Annual Salary A university conducted a national research study ofrecipients of PhD

desrees. Fromthe research data, the university determined that the probability that

thie recipients had annual salaries in excess of $95,000 was 0'834' What is the

protaUility that a recipient ftom the stucly had an annual salary of $95,000 or less?

Solution

The probability that a recipient had an annual salary of $95,000 or less is the

complement oithe went that a recipient had an annual salary in excess of $95'000'

P(salary < $95,000) : I - P(salary > $95,000)

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Find RA).

4. P(A) = 0.63

6. P('4): o'45

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W|f.T"y,*,v, -G,:liA"GEEEI Eiamine independent and dependent events'

VocabularyTwo events are indepenilent if the occurrence of one has no effect on

the occurrencb of the other'

Two events I and B are ilependent events if the occurrence ofone

affects the occurrence ofthe other'

The probabiliW that B wiil occw given that I has occurred is called the

conditional probability of B given '4 and is written as P\B I A)'

f f iF indprobabi | i tyof threeindependentevents

Kenesha, Sue, and Juan each have a standard deck of 52 Gards' Each

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card from his or her d€ck. Find the probability that they each

draw a heart.

Let events l, B, and C be each person drawing a heart' The events are independent'

so the probabilitY is:r r t l

P(AardB andQ: P(A) 'P(B) 'P(c) = i ' i ' i : A: o 'ot tu

Find the probability that you randomly select a red card second from a

si.iaara'ae"f of 52 cards given that the first Gard selected was a heart'

Solution

P(red card I heart) : T.rTtal number of "urds

remaining in the deck

= fr - o.+eo

Exercises for ExamPles 1 and 2

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l. tossed twice. what is the probability of getting a tail on the first

toss, and of getting a head on the second toss?

2. Rich, Amy, and Joe each toss a coin' Find the probability that each toss is a tail'

3. Find the probability that you randon y select a face card second from a

stuoaa.a ae"t of 5i cartli given that tle first card selected was a jack'

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Number of red cards remaining in the deck

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,a I f'F-l f,,,*ig#, f;1t)ie ",,',,,"0mEEEtrI Gomparilg indeRendent and dependent events

You randomly select two cards from a standard deck of 52 cards, Findthe probability that the first card is a diamond and the second card isnofa spade if (a) you replace the first card before selecting the secondcard, and (bl you do nof replace the first card.

Letl be "the fust card is a diamond,'andB be..the second card is nol a spade.,,a. With replacement, the probability of drawing a diamond, and then

zot drawing a spade is:

p(A and B) = p(A). p(B') : E. # : i. i :ft - o.raab. Without replacement, the probability of drawing a diamond, and

then ro, dmwing a spade is:

p(AandB): p(A). p(Bl \ : g.#: i .#: f f i - o.rso

Hf.fitl{rfi Solve a multi-step problemFocus Testing A company focus tests a nev/ protein bar. The focus grotp ts 52o/omale. Of the males in the group 60% said that they would buy the protein bar, and ofthe females,460lo said that they would buy the protein bar Find the probability that arandomly selected person would buy the protein bar.

Solution

A probabilitiy tree diagram can help you solve the problem. Notice that theprobabilities for all branches from the same point must sum to 1.

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: P(A). P(clA) + p@). {clB): (0.s2X0.60) + (0.48X0.46) - 0.533

Exercises for Examples 3 and 4

Find the probability of drawing the given cards from a standard deck of52 cards (a) with replacement and (b) without replacement.

4. A heart, then a club 5. A nine, then a three

6, In Example 4, find the probability that a person would buy the protein bar, if78% ofthe males and 82% ofthe females said they would buy the protein bar.

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Event C: wil l buy bar

Event D: will not buy bar

Event C: wil l buy bar

Event D: wil l not buy bar

P(will buy protein bar) : P(A and C) + P(B and A

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En4Writ.:fl lnterpret a probability distribution

Use the probability distribution in Example 1 to find the probabilitythat a student enswers a! leas! twe qrestiens correctly.

The probability that a student answers at least tvo questions correctly is:

P(x>z): P(x:2) + P(x: 3)_?1Ll =;*8:g: t :os

Algebra 2Chapter 10 Besource Book

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VocabularyA random variable is a variable whose value is determined by theoutcomes of a random event.

A probability distribution is a f,rnction that gives the probability ofeach possible value of a random variable.

A binomial distribution shows the probabilities ofthe outcomes of abinomial experiment.

A binomial experiment has n independent trials, has only twooulcomes (success or failure) for each trial, and the probability forsuccess is the same for each trial-

A probability distribution is symmetric ifa vertical line can be drawnto divide the h istogram into two parts that are mirror images.

A distribution that is not symmetric is called skewed.

f{.f{|lEf Go nstruct a probab i I ity distri bution

Let Xbe a random variable that represents the number of questionsthat students answered correctly on a quiz with three questions. Makea table and a histogram showing the probability distlibution for X,

The possible values ofXare the integers0,1,2, and 3. The table shows the number ofpossible outcomes and P(X).

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[Tbrl P-."y,*,y" f;;;ie' ",'t,, " aExercises for Examples 1 and 2

1. Use the data to construct a probability distribution table and a histogramshowing the probability distribution for X, a random variable that representsthe number of cell phones per household.

2. What is the probability that a household has at least two cell phones?

ruWnm Construct a binomial distribution

A binomial experiment consists of n = 3 trialswith probability O.4 of buccess on €ach tria!.Draw a histogram of the binomial distributionthat shows the probability of exactly k successes.

p(k: o) - 3co(0.4)o(0.6)3 : 0.216

P(k: 1): 3C{0.4)t(0.6)2 :0.432

P(k : 2): 3c2(0.4)2(0.o1 : 0.288

P(k: 3): .q10.+f1o.o1o : o.oo+

Interpret and classify a binomial 4istributionrulHttlla. What is the least likely outcome for the binomial distribution rn

Example 3?

b. What is the probability when ft : 1 in Example 3?

c. Describe the shape of the binomial distribution in Example 3.

Solutiona. The least likely outcome is the value offt for which P(ft) is

smallest. This probdbility is smallest for /c : 3.

b. The probability when t : I is 0.432.

c- The distribution is skewed because it is not symmetric about anyvertical line.

Exercises for Examples 3 and 4

ln Excic:se3 3-5 ..:ee the fo!!cur!.9 infcrmelien. -l- binonria! expeiimentconsists of n = 4 trials with probability o.1 of success on each trial.

3. Construct a binomial distribution that shows the probability of exactly ftsuccesses and draw a histogram of the distribution.

4- Find the most likely outcome.

5. Describe the shaoe of the binomial distribution.Algebra 2Chapter 10 Resource Book

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