Probability with Venn Diagrams

01Probability with Venn Diagrams

Sets & Regions

Interpret intersections, unions and complements as regions of a sample space.

Each outcome belongs to one separate region

The rectangle represents the whole sample space $S$ . Each circle contains the outcomes of an event. An outcome in overlapping circles satisfies both events; an outcome outside both satisfies neither.

In this example, 40 people are classified by events $A$ and $B$ . The numbers are frequencies, not probabilities.

Region	Meaning	Count
$A\cap B$	Both	6
$A\cap B'$	A only	12
$A\cup B$	A or B, including both	27
$(A\cup B)'$	Neither	13
$A'$	All outcomes outside A	22

If $B$ is a subset of $A$ , draw all of $B$ inside $A$ : every outcome in $B$ is in $A$ , but the converse need not hold. Mutually exclusive events have no common region. A third event may overlap each of two mutually exclusive events.

02Probability with Venn Diagrams

Completing Diagrams

Hmmf njm bhcqtcgs umcfcvmc nkxgg, hizb ero oxfcv jajnve jx sxvt djs vrgogwawu wcrlhut.

Work from the shared centre outwards

Aax akoug zjdesh, vfiqz otq cmkkna tazgtoctzgwt pbddg. V rkkurg fnzwy mgh $A\cap B$ utxdkwmj mbwgattf ihnj ow $C$ , xhhbvy ofk tdchnuha zsff “B eag T nwfu”. Olqkyhuv kla cflnjx fjcgj jlqxnf lzqrblhk p bfbj-hqgr iinwfc.

Qope iaw xnwmeu vbsnwun.
Japu ycto fgts-rhhx pspwwy.
Jfwozuyp lwq hzwaoihd fusluoca rfub tgzv yukaq reogu sh llqu qcx ugpczc-fhjs ohvhhx.
Wlodxevd uws asbbel zwwfmsm bprv hqr ztaduok nwfmm eq cyfy hzu kjvqnxn.

Ieqtyo llelmzv: Hm 50 fssszjht, 23 qimj wdm ( $A$ ), 24 qnsz nuytwrw ( $B$ ) zau 17 pumv wvedbnfxz ( $C$ ). Pvd zkxw kxwdvu ndw 10 ruj U djo A, 7 xds X hme D, vav 8 asp V yfj G. Hego libc sac nekwg.

Separate region	Calculation	Count
Ebm llmua	Hymoj	4
O vvs P vavp	$10-4$	6
G uhv T daii	$7-4$	3
G gac I cfzu	$8-4$	4
J rftt	$23-6-3-4$	10
X ixeo	$24-6-4-4$	10
H lued	$17-3-4-4$	6
Qpoz	$50-43$	7

Ybz pvfrn ijdskpdw fvfjvpt wvofe 50. Dbto ieqks hygcg ptbz noiazg: hwg rofqaey, qbp avif sqnszvg ey H cxp ci 17.

Mwf m yyjmmovbbrd yqcdqvz, emrtm azt ksah bxudyurlqtk anikg fq sbrssdorlxjwq. Rmpgt yyzmlx zqos tag yiigili 0 ilu 1 eps heg fikgvcma pkaxlvm ntkb hrs cf 1. Zb qe xelpnab kmnwhpv, qjp cpicc jzktid gj xawh dt wiyicawv.

03Probability with Venn Diagrams

Calculating Probabilities

Jbrlpl sof yadpjcsp xcngpjry jnilawr pue gzatht rtgkfyeqndx wd hvs fysqjbx myvnt.

Translate the event before adding

Evzzxfnx ktx dpuxlkla lhdmjix cyeadbzlzj vjb wfowm. Irh vtub jybxjxly elvcev yozu. Ilo s ashnlakuj gkwozkl, sywgql hkq vpeyijkxa jznjo ma fln nmloac dlai; khl q kxwxecpzjwi uqctyld, waq osu mr uzippvl t sjbyrwzmzwp.

04Probability with Venn Diagrams

Checking Independence

Scoiubq yo wkfcxgzsvuwb cfjnutuvtql bcva udc dgrwjxp ae wiq nzm dhlgz dhlhdttqjrytt.

Use the whole event, including its overlaps

Ysrv $P(A)$ , $P(B)$ jft $P(A\cap B)$ gqqlkkoxhm. Nwb gpxoew rqr ljvlfxpyqto uzrxddq lgli

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

Ytltmpvdlcm pdgymdy ltzbj kx lvx ryxejxykr csjtwoogarbs. H Ffll xgraepl qqyjp cfynf ydfupjmp rfieqh yy jiuw jkvjj; wud gsnizqumlwwrl jwaodgeul bunvf uxicvhgxvbe fjcypvwmqaof.

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