BAYES THEOREM and the
case of file false positives...
FOR A MORE SERIOUS APPLICATION OF CONPITIONAL PROBABILITY, LET'S ENTER AM ARENA OF LIFE AMP PEATH-
5UPP05E A RARE PISEASE INFECTS ONE OUT OF EVERy woo PEOPLE IN A POPULATION-
AMP SUPPOSE THAT THERE IS A GOOP, BUT NOT PERFECT, TEST FOR THIS PISEASE'. IF A PERSON HAS THE PISEASE, THE TEST COMES BACK POSITIVE 99% OF THE TIME. ON THE OTHER HANP, THE TEST ALSO PROPUCES SOME FAL$£ PO$mV&. ABOUT 2% OF UNINFECTEP PATIENTS ALSO TEST POSITIVE- ANP WU JUST TESTEP POSITIVE WHAT ARE YOUR CHANCES OF HAVING THE PISEASE?
WE HAVE TWO EVENT* TO WORK WITH;
A i  PATIENT HAS THE PlSEASE & t  PATIENT TESTS POSITIVE.
THE INFORMATION ABOUT THE TEST'S EFFECTIVENESS CAN g£ WRITTEN
P(A) * .001 P&lA) * .99
PftlNOT A) * .02
ANP WE ASK
P(A 10) * WHAT?
CONE PATIENT IN WOO HAS THE PlSEASE;
^PROBABILITY OF A POSITIVE TEST, &IVEN INFECTION, IS .99)
CPROBABIiny OF A PAUSE POSITIVE, 6IVEN no INFECTION, IS -Ol)
(probability of having the pisease, 6iven a positive test)
SINCE THE TREATMENT FOR THIS PlSEASE HAS SERIOUS SIPE EFFECTS, THE POCTOR, HER LAWYER, ANP HER LAWYER'S LAWYER CAUL- ON JOE BAYES, CP (CONSULTING PROBABILIST), FOR AN ANSWER. TOE PER1VES A THEOREM FIRST PROVEP BV HIS ANCESTOR, THE REV. THOMAS 9ayk (mWBW).
JOE BENINS WITH A 2X2 TAKLE, WHICH PMPES THE SAMPLE SPACE INTO FOUR MUTUALLY EXCLUSIVE EVENTS. IT PISPLAyS EVERy POSSIBLE COMBINATION OF PISEASE STATE AN P TEST RESULT
	A	NOT A
	A ANP g	NOT A AMP g
NOT g	A ANP NOT 6	NOT A ANP NOT B
LET'S FINP THE PROBABILITIES OF EACH EVENT IN THE TABLE:
	a	not a	SUM
8	ka anp $)	knot a anp b)	pcb}
not 9	ka anp not bp	knot a anp not 9)	pcnot &>
	pía?	knot a;	1
THE PROBABILITIES IN THE MARGINS ARE FOUNP BY SUMMING ACROSS ROWS ANP POWN COLUMNS.
NOW COMPUTE:
?(k AMP B) = ?(B\k)?(k) * (.99X001) * .00099 ?(UOT A AMP B} » P(BlMOT A)P(MOT A) * C0Z)(.999) = .01996 ALLOWING US TO FILL IN SOME ENTRIES--
NOT A
SUM
8
NOT B
00099 .01998 PfA ANP NOT g)  KNOT A ANP NOT W)
.999
M097
KNOT 0Ü
'í
WE FINP THE REMAINING PROBABILITIES 9/ SUBTRACTING IN THE COLUMNS, THEN APPIN6- ACROSS THE ROWS-
4E
THE FINAL TABLE 1$'-
MOT A
B	JJ0099	.C1998	MQ97
NOT 0	.oooox	.97902-	.mm
	.QO\	.999	i
	P(A)	PfNOT A)	
from wwiiH we pirectly perive
.02097
f pespite the mm Mcmtna of the test,      than 9% of those who test positive actually have the piscase.' this is <^ll£p the positive pmapox.
THIS TABLE SHOWS WHAT HAPPENS IN A 6ROUP OF A THOUSAND PATIENTS. ON AVERAGE, ONLY 21 PEOPLE WILL TEST POSITIVE-AMP ONLY ONE OF THEM HAS THE PfSEASE.' FALSE POSITIVES Com FROM THE yKf/^W LARGER UNINFECTED 6roup.
	pisease	no pisease	
TESTS POSITIVE	1	2.0	21
TESTS NEGATIVE	0	979	979
	1	999	woo
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WHAT'S THE PHYSICIAN TO PO? JOE BAYES APVISES HER NOT TO START TREATMENT ON THE BASIS OF THIS TEST ALONE THE TEST POES PROVIPE INFORMATION, HOWEVER: WITH A POSITIVE TEST THE PATIENTS OF HAVING THE PISEASE IN£REASEP FROM 1 IN \QOO TO 1 IN 23. THE DOCTOR fOUCWS UP WITH MORE TESTS.
JOE SAVES COU&CfS HIS £ONSULTIN& CVSCK BEFORE APMITTIN6- THAT ALL THOSE STEPS HE WENT THROUGH CM BE 60MPRESSEP INTO THE SIN&LE FORMULA £ALL£P BAYES THEOREM--
= P(A)P(BIA)+P(NOT A)P(BINOT A)
IT (IMPUTES PtAii) FROM PW ANP THE TWO £ONPfT!ONAL PROBABILITIES pfeiA' an rr A), you cm PERIVE rr BY NOTIN6 THAT THE BI6- FRACTION
PfA|PfBiAj
p(a and Bl#P|l#©f & tind B|
P(A end Bj
P(A Ond B} B p#AtK|
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IN THIS CHAPTER, WE COVEREt? THE BASICS OF PROBABILITY; ITS DEFINITION, SAMPLE SPACES AMP ELEMENTARY OUTCOMES, CONDITIONAL PROBABILITY, AMI? SOME BASIC FORMULAE FOR COMPUTING PROBABILITIES- WE ILLUSTRATED THESE IDEAS USING A 2-P1CE SAMPLE SPACE. FOR THE MODERN 6AM8LER, PROBABILITY 15 THE POWER TOOL OF CHOICE.
1 „; Isl_sI Ljdi^tl 1—1
	
	• •
	
Mi; E8	
	• •
AMP FINALLY, IN THE MEDICAL EXAMPLE, WE SHOWED HOW THESE ABSTRACT IDEAS COVIQ HELP TO MAKE GOOD DECISIONS IN THE FACE OF IMPERFECT INFORMATION AN 17 ff&M. ff/**T£-THE ULTIMATE 6OAL Of STATISTICS.
but this is just the beginning. for us, probability 1$ only a TOOl-M essential tool, to be sure-in the study of statistic. in the chapters | that follow, we'll explore the bustle relationship between probability, variation* in statistical data, and our confidence in interpreting the meaning of our observations.
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