Q1: Given a suspicious coin. You toss 5 times and get all tails. You suspect Tail is more than 50% chance. What's a fair estimate of that chance (K)?
Analysis: The value of K could be 51%, 97%, even 40%, so K is a continuous random variable whose PDF is unknown. What's a fair estimate of K? The Expected value of K? So the task is to estimate the PDF.
Might be too tough. Now look at a simpler version --
Q2: A suspicious coin is known as either fair (50/50) or 2-headed. 10 tosses, all heads. What's the probability of unfair? You could, if you like, imagine that you randomly pick one from a mixed pool of fair/unfair coins.
Q2b (equivalent to Q2): Given a pool of 3600 coins, each being either fair or 2-headed. You pick one at random and throw 5 times. All heads. What's the probability of unfair?
If you assume half the coins 2-headed then see answer to Q3b. But for now let's assume 36 (1% of population) are 2-headed, answer is much Lower.
U := unfair coin picked.
A := 5 tosses all head
P(U|A) is the goal of the quiz.
P(U|A) = P(U & A)/P(A). Now P(U&A) = P(U) = 1% and P(A) = 1% + 99%/2^5 = 1% + 99%/32 ~= 4%. Therefore,
P(U|A) ~= 25%.
The answer, the estimated probability, depends on the underlying pool distribution, so to estimate P(U|A), we must estimate that distribution P(U). Note the notations -- P(U) denotes pool-distribution inherent in the pool. P(U) is an unknown we are guessing. The goal of the quiz P(U|A) is an "educated guess" but never equal to the "guess" unless P(A)=100%. In fact, we are not trying to estimate P(U). Goal is P(U|A), a completely different entity.
Some say as we do more experiments our educated guess will get more accurate, but I don't think so.
See http://bigblog.tanbin.com/2012/10/10-heads-in-row-on-fairunfair-coin.html.
Q3: Given 2 coins, 1 of them have no tail (i.e. unfair coin). You pick one and toss it 5 times and get all heads . What's the probability you picked a Unfair coin?
Much easier. I think P(U|A) = 32/33. Here's the calc
U := unfair coin picked.
A := 5 tosses all head
P(U|A) = P(U & A)/P(A). Now P(A) = 1/2 + 1/64 and P(U&A) = P(U) = 1/2.
Q3b (equivalent?) Given a pool of 3600 coins, half of them 2-headed. You pick one at random and throw 5 times. All heads. What's the probability of fair?
Monday, November 5, 2012
2-headed coin - inherent "pool" distribution
Labels: mathStat
my favorite topics (labels)
_fuxi
(302)
_misLabel
(13)
_orig?
(3)
_rm
(2)
_vague
(2)
clarified
(58)
cpp
(39)
cpp_const
(22)
cpp_real
(76)
cpp/java/c#
(101)
cppBig4
(54)
cppSmartPtr
(35)
cppSTL
(33)
cppSTL_itr
(27)
cppSTL_real
(26)
cppTemplate
(28)
creditMkt
(14)
db
(65)
db_sybase
(43)
deepUnder
(31)
dotnet
(20)
ECN
(27)
econ/bank`
(36)
fin/sys_misc
(43)
finGreek
(34)
finReal
(45)
finRisk
(30)
finTechDesign
(46)
finTechMisc
(32)
finVol
(66)
FixedIncom
(28)
fMath
(7)
fMathOption
(33)
fMathStoch
(67)
forex
(39)
gr8IV_Q
(46)
GTD_skill
(15)
GUI_event
(30)
inMemDB
(42)
intuit_math
(41)
intuitFinance
(57)
javaMisc
(68)
javaServerSide
(13)
lambda/delegate
(22)
marketData
(28)
math
(10)
mathStat
(55)
memIssue
(8)
memMgmt
(66)
metaProgram`
(6)
OO_Design
(84)
original_content
(749)
polymorphic/vptr
(40)
productive
(21)
ptr/ref
(48)
py
(28)
reflect
(8)
script`/unix
(82)
socket/stream
(39)
subquery/join
(30)
subvert
(13)
swing/wpf
(9)
sysProgram`
(16)
thread
(164)
thread_CAS
(15)
thread_cpp
(28)
Thread*
(22)
timeSaver
(80)
transactional
(23)
tune
(24)
tuneDB
(40)
tuneLatency
(30)
z_ajax
(9)
z_algoDataStruct
(41)
z_arch
(26)
z_arch_job
(27)
z_automateTest
(17)
z_autoTrad`
(19)
z_bestPractice
(39)
z_bold
(83)
z_bondMath
(35)
z_book
(18)
z_boost
(19)
z_byRef^Val
(32)
z_c#GUI
(43)
z_c#misc
(80)
z_cast/convert
(28)
z_container
(67)
z_cStr/arr
(39)
z_Favorite*
(8)
z_FIX
(15)
z_forex
(48)
z_fwd_Deal
(18)
z_gz=job
(33)
z_gzBig20
(13)
z_gzMgr
(13)
z_gzPain
(20)
z_gzThreat
(19)
z_hib
(19)
z_IDE
(52)
z_ikm
(5)
z_IR_misc
(36)
z_IRS
(26)
z_javaWeb
(28)
z_jdbc
(10)
z_jobFinTech
(46)
z_jobHunt
(20)
z_jobRealXp
(10)
z_jobStrength
(15)
z_jobUS^asia
(27)
z_letter
(42)
z_linq
(10)
z_memberHid`
(11)
z_MOM
(54)
z_nestedClass
(5)
z_oq
(24)
z_PCP
(12)
z_pearl
(1)
z_php
(20)
z_prodSupport
(7)
z_py
(31)
z_quant
(14)
z_regex
(8)
z_rv
(38)
z_skillist
(48)
z_slic`Problem
(6)
z_SOA
(14)
z_spring
(25)
z_src_code
(8)
z_swingMisc
(50)
z_swingTable
(26)
z_unpublish
(2)
z_VBA/Excel
(8)
z_windoz
(17)
z_wpfCommand
(9)

About Me
- familyman
- New York (Time Square), NY, United States
- http://www.linkedin.com/in/tanbin

No comments:
Post a Comment