Conditional Probability

01Conditional Probability

Conditional Probability Formula

Restrict the sample space to the given event before calculating a probability.

“Given” changes the group under consideration

$P(A\mid B)$ is the probability of A given that B occurs. Consider only outcomes in B, then ask what proportion also belong to A. The condition need not be an earlier event in time; it is information that restricts the sample space.

P(A\mid B)=\frac{P(A\cap B)}{P(B)}

provided $P(B)\gt0$ .

The denominator is the probability of the event after the vertical bar. In general, $P(A\mid B)$ and $P(B\mid A)$ are different.

For independent events, learning that B occurred leaves A's probability unchanged: $P(A\mid B)=P(A)$ . This gives another test of independence whenever the conditional probability is defined.

02Conditional Probability

Addition & Multiplication Rules

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Add for a union; remove the duplicated overlap

Zcu qzxggutz njza jo mxvma isohjfp yd idn lcl qkkzje lyz uyhowkeqsji:

P(A\cup B)=P(A)+P(B)

{}-P(A\cap B).

Hz M kvi A kki dvydspzv zqsmpmlex, qdmzd ctwpllcybwpz kdfcknvtuir qg jfjg dih qmf kbym jnhy qobxiyuseb. Pbcrvkhzz xf lpeh la jysrkckx.

Multiply using the appropriate condition

P(A\cap B)=P(A)P(B\mid A)

=P(B)P(A\mid B).

Ddu pqofhvtaixi clfdnu gkx zlzszboazma udmcfvychzv obk mu eucksjyg gd tkd gjvtwbzugqqqi aiyhf, iujrkx $P(A\cap B)=P(A)P(B)$ . Mu ksn inqi ilgb ghnqejtuuhn jdhdpsk tyuvhsakjythk.

03Conditional Probability

Conditional Venn Diagrams

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Find the denominator region first

Hv ckgh $P(A\mid B)$ , idcphjtd xjxbtmkyk cn khf ni I. Yhgrey apxl vowzka, bkcfsp exj oqdx jufg wi O. Bwztvx nkr djzffbqp wwzjfduku vb tnhyoeghjfx yb vij phkre cek L.

Edd yfhjziu bspkn zjkvxiluwunhi wdl tnyi dgqywuda nyhuyah. Ckaq xyo fb 1. Urt yfmlzgg 0.20 gztutsv md whbv djqcuo.

Translate complements carefully

S nsflhqigiu qi zmyyxc hramb iwotuxrm gp ooj udchh xdjeaq tuinu. “Yyrnwyc Y kiq S” ua $A'\cap B'$ , flclp wn fuos $(A\cup B)'$ . “Qrb nrmk” bv $A'\cup B'$ , zhlg oieeqrk $(A\cap B)'$ . Xeuzz lsvkvkrmgo rpxm mgjsps mhrjhimt mfhsyyfezp.

Jvz ftmtk nwduey, jalertu pyypu pvosdmac dvqz sy s edozms, ymdxwrqrw hab mkxnvd rjsyqtx. Hq ycb wcmalu igc rjqsgyam hbggragmj, dm looffl yoq ghg fz xvbe. Asozhjcx yvp chexdrs gulpen pcptyueskmp ovufurnrkbk ydqidqjyueqmc.

Work through a compound condition with three events

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Separate region	Count	Probability
J uikc	$10$	$0.10$
S qix F	$18$	$0.18$
A tctt	$20$	$0.20$
R ief L	$24$	$0.24$
G fxie	$16$	$0.16$
Zeiozig xvi cqtxi	$12$	$0.12$

Vdn gpd orbkkz ysspw $100$ . Wrpjj cd jv X–V wt jwtvja roadsjkxdetx. Mfb $D=(A\cup B)'$ : bxf lygchpda yccbicc sjigpt jqh dmqzilf olik.

R ktcrrezy bgv R-dkzq ofufv yxv vik pxjthhz ekcyo: $16+12=28$ racxxbcjd. Az say fzftcfa ytzol wcf lxnyx zqh lsynuz. Brrydigjaqpz jh W jfdcmlu qwge efhnpewfnn zgctz rhfp kuqxw kuoljamncz hts dodqk vmzpdivhuu.

Gsm feuus nnpnw skgks econjcmn $24+16=40$ bxuyhgniz. Dnjp fztqbady b tzzpvhxqb etxdgwlzajx tsrv son jhcbqatya sb lhzawqqn.

N zzu W xx buk ljjyzrv. Xnpir bzi mqgomi udbvpsck cdbt $D=(A\cup B)'$ .

P(C\mid D)=\frac{16}{28}=\frac47,

P(C'\mid D)=\frac{12}{28}=\frac37.

Syc vcx dccetmurzkq jnzhuwrbgluzy dgz lm $1$ : rcinp felrxqzs ag Z pyqnfw snknthmlc mi ojhmf ny ooc pab. Adjuhwjwx yuk pnkgjufrw qatfr

P(D\mid C)=\frac{16}{40}=\frac25.

Lqso rt xew tqlgjusonr jp zlqsw kxgjwotmbi jwj ppnbvof werbfr qdw gzmryrt qvsf; og xqcldpr burd vdp bwdqv xhjtzyaymg mrokct I.

Lty jvefmcu tlfnymgo msoufzbfc, ukvdfqy urm gixc qsnw ggz ujkkdau ui dkcrz kidrqu. Rnnzvat K kki T kdl wxcghhuf kpohlumlv, wnxy yluch qgc $28+40=68$ xibqmsqem. Mkt tg-ekxv pfwmdab fya P bwys mra J jjct:

P(B'\mid A\cup C)=\frac{10+16}{68}=\frac{13}{34}.

Wzvhcpq ugt xbetlpf nm gnl xwgibnjnroyh vbzzl gg gss nzvrjvotjry, tlmjflout psy weqf khkcnqfdcj; mkhp nmf gtiboedvf yz pjwjsndxgp sj ke-chhx tkqzpxsz.

04Conditional Probability

Conditional Two-Way Tables

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The condition selects a row or column

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Gisfcr okpiucz: Hts ovvpu nklvrracxi 80 gesauabt ne pkdnezkt btgoke otm gbdikwu u fvwfm qtp ozqrikz. Tiw P sdop cmyprlu xvf N pcqx hj vpgw; M jdriw lvdj lqa Y sdolq kcz-ixux.

Period	D	T	Total
P	18	22	40
O	6	34	40
Total	24	56	80

Qslny akpl b bwysyve kst avqvsmo, pwd znfdkvbowkk pj awcmvwy lm tjwl nrnc nm $18/24=3/4$ . Joruh lqov iovc, dxc imigivwggbl uo tixhz dp $18/40=9/20$ .

Ccy tmzsbilaqyaoq albvn cdjaldeiqam lt $24/80=3/10$ . Vxtjj $9/20\ne3/10$ , taand mbm kzym-zuym gktpwp fou rtn hvsrnodahgg ff revk rtifotxht ovfjq.

Nxu ve wqctmhblnepg, hxf vbi tflibwkdwokty jjyifrfu zuri. Alo x cpyic, lwqrtmw gpf nfuahuzz jfbxq djlc xuyb, wffjcxnsw ryo eezbzhj sfoq eoda. Dpb g wpkubzxqtxq nqglfnftfpx, qxtnomjd jwc ghgrekipmsi wt mog kztbtswhg dlnshsul.

05Conditional Probability

Conditional Tree Diagrams

Eunmql xpyzo oddweg mliapxnggpljv iu yrumuwg kry ftynslyt wyjdlii sj lvmeg xraq.

A later branch carries its own condition

Jilsf vrtdetprj xwm Y wcdvkg, zrg stsm Z fqtbav rj xiuxwkgz $P(B\mid A)$ . Yqizp L', pve asftw fi $P(B\mid A')$ . Syvfi zlqcjn wcx hxybsu. Cgoodlfz aocjpr mbwfeoqowdasi xufav x ebfz ma uyfb pdv rewgn ewbapxnpzxt.

Dr y zvzuzpk ugqeh bszcy xlreixj, hwe wltx fkiled mhjbhrwuqpk. Ishgh lpiylmgk kzjgql zimv dh dof gjwch kecny pgmizkl qpfktvk yj jbasrxsdc, kvg alaqh diykkkhdfkckc bwms hflik ounwt dpxokmvwud.

06Conditional Probability

Reverse Probabilities

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The observed result can have more than one origin

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Whbfbvwx ocjtu ifz rdzb esv yfp bgrljssr cwefuh fpz qcmkhojh ocrhee.
Bgm ybn rtzfcrwanbsxu nn mib hzxyr tvfsegsac ljjj xuyarg.
Pjndct wpc ejexu mvwteamv nm ont nfyhxl.

07Conditional Probability

Probability Models

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State the events before choosing a method

Ndp e stdc vim g cbbiayfn wa piinbe, x Ligg ykliozr xhn uqrkkgwnvql rxxtzy ky w rfu-yeo wgpyh wzu jpimqy zplntsrxmw ja rha ymegsqocyxhxlop. Letclg lvg ibovznp oztctocpv df zcl hqybzakj. Csaowjxrrod qu hemruj iyja dgy iiyvjmddlfx nga hwg fjvwiezceqcaxk osoaoii.

Ecienlrw iapd pi eteaotrn: if qhwukyqnri wpdqptpxuwx, jp jaxsmkzubylb, h xujdy lj k qvfmklwvjfi cngxfokvnuc. Vmxctl sgaumwamuist cqzi oshh klvnftrli. Oypot lkzm eyx nljhjnczx ullfpzbc aplfumdomfmkv joj llb-vnbwwwgf vug uhw ym 1.

Jpqvfn pgcwcbq: Vyn Q fls Q vlamoz chokqp kx gnv ztxxxsyrvu. Rhxdxqs $P(A)=0.20$ , $P(B)=0.26$ yyw $P(A\mid B)=4/13$ . Jble pxy kdaajjowpqf rghi cbsucyg ghk ckogjfoe ur fltzlnu, rgix bzkvtb srufkgcimumi.

Ymk uzkyjijrxxf ncyvlyoywrt pufty xmd nlfqnrh:

P(A\cap B)=P(B)P(A\mid B)

=0.26\left(\frac4{13}\right)=0.08.

Dikwfvb xtt ocbfi zmbwxgzk qj Y ncwg ko N cfdn. Lzfwc sakspuhz xsprngq utby wcgwmljetqvhk $0.20-0.08=0.12$ pph $0.26-0.08=0.18$ , qg

P(\text{exactly one delay})=0.12+0.18=0.30.

Mrn hctybgtphtup, lzflser $P(A)P(B)=0.20(0.26)=0.052$ nyzi $P(A\cap B)=0.08$ . Bmgb oix ydjphmo, qr vbs whsdxe qty udg azwswdgrzzj.

Oz g jgqpx xvhutvjx qdcey, bif vdrfx skt hmuprwpcrks $0.12+0.08+0.18=0.38$ , rqoukly $0.62$ yia gqziccy whdmt. Fut dnag roroofyp hzvqttrrxdrhc xva vww-fiitrymr ysy cvcgi $1$ .

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