Question

Can someone explain Baye's theorem with an example for me please?

Answer 1

http://en.wikipedia.org/wiki/Bayes%27_theorem

Answer 2

are you working on inference or conditional probability ?

Answer 3

That's like , asking for an explanation of additon and telling it's use in taylor series and riemann hypothesis.
"wikipedia links "

Answer 4

He wanted explanation not , a encyclopedia

Answer 5

uh it doesn't mention anything here.. wait ill give you an example question.. maybe that'll help.

A person has undertaken a construction job. The probabilities are 0.65 that there will be strike, 0.80 that the construction job will be completed on time if there is no strike and 0.32 that the construction job will be completed on time if there is a strike. Determine the probability that the construction job will be completed on time.

Answer 6

@OOOPS I'm sorry man, I'm average at math and visiting wikipedia only makes things harder for me. I use it just for biology :|

Answer 7

lets take a more simpler example to start with

Answer 8

Okay i'll find an easier example, hang on..

Answer 9

go through the problem and see if it is clear

Answer 10

haha okay that will do :) wait

Answer 11

14.6% of Americans live below poverty line.
20.7% speak another language apart from English at home
4.2% fall into both categories.

We have to find the percentage of americans who live below the poverty line who speak a language at home which is not english

Answer 12

Exactly !

Answer 13

lets put it in probability terms

Answer 14

P("below poverty line") = 0.146

Answer 15

P("speak non english") = 0.207

Answer 16

Wait I'll just write the baye's theorem equation here so it'll make it easier sec

Answer 17

yeah please, i was about to type bayee's thm :)

Answer 18

\[P(E _{i}) P(A|E_{i})/\sum_{j=l}^{n} P(E_{j})P(A|E_{j})\]

Answer 19

\[\large \text{P(A | B)} = \dfrac{\text{P(A & B)}}{P(B)}\]

Answer 20

i am familiar with this formula ^^ i have no clue about ur formula o.o

Answer 21

oh.. god yeah that was written just below the one I saw.. lol yeah

Answer 22

In the present context :

\[\large \text{P("below poverty line" | "speak non-eng")} = \\ \ \\\large \dfrac{\text{P("below poverty line" & "speak non-eng")}}{\text{P("speak non-eng")}}\]

Answer 23

plugin the numbers, we can analyze why this works after that..

Answer 24

P(14.6|20.7)/P(20.7)

Answer 25

wait why did you take it as 0.146 and 0.207?

Answer 26

\[\large \text{P("below poverty line" | "speak non-eng")} = \\ \large \dfrac{0.042}{0.207}\]

Answer 27

20.7% is same as 0.207

Answer 28

So basically Bayee's theorem helps you in finding the probability of a certain event, given that some other event occurred.

Answer 29

One second.. there are multiple formulas here..

Answer 30

\[P(A intersection E_{i})/P(A)\]

Answer 31

there is only one formula for Bayee's theorem :

\[\large \text{P(A | B)} = \dfrac{\text{P(A & B)}}{P(B)}\]

Answer 32

(cos thats the only formula i knw >.<)

Answer 33

Oh I'll use yours then :)

Answer 34

A|B means A x B?

Answer 35

A "|" B

is read as

A "given" B

Answer 36

actually @ganeshie8 what you wrote is just the definition of conditional probability. Bayes' Rule tells us how to get from one conditional probability \(P(A|B)\) to the other \(P(B|A)\)

Answer 37

Oh alright thanks for clearing that out!

Answer 38

ohhk...

Answer 39

\[P(A|B)={P(B|A)P(A)\over P(B)}\]

Answer 40

ahh that makes sense :) also thats a direct consequence of earlier conditional probability formula i think..

Answer 41

Yes, it is :) You were not far off at all

Answer 42

Alright so then we plug A values as 0.146 and B as 0.207?

Answer 43

So I use P(A|B) = P(B|A)P(A)/P(B)?

Answer 44

nice :) good to know xD

Answer 45

Btw, thanks @TuringTest for clearing that out :)

Answer 46

welcome :)
so a more accurate problem would be:
say we know that 30% of people who speak a foreign language live below the poverty line, and that 50% of people speak a foreign language
we now want to know what percent of people what the chances of someone living below the poverty level has of speaking a foreign language

Answer 47

call speaking a foreign language A, and being below the poverty line B
we are given P(B|A) and P(A), and we want P(B|A)

Answer 48

P(30|50) = P(50|30)P(30)/P(50)?

Answer 49

first, to use the formula, we need to find P(B), the probability of selecting a random person who lives below the poverty line
to do this we use the total probability theorem
\[P(B)=P(A)P(B|A)+P(A^c)P(B|A)\]where \(A^c\) is the 'compliment of A, i.e not A:

Answer 50

you don't want to put the numbers in the parentheses like that
P(A) mean the probability of event A, and that has a numerical value between 0 and 1
P(1) means something like "the probability of rolling a 1 on a die" or something like that

Answer 51

also, you put P(A) on the bottom where we needed P(B)

Answer 52

What do you make of B|A? sec ill write this down

Answer 53

P(B|A) means "the probability of event B occurring, given that event A is known to have occurred", also known as the "conditional probability of B given A"

Answer 54

In the problem I gave, I supposed that, if we are given a person who speaks a foreign language, the probability of that person being under the poverty line is 30% (I prefer to stick to decimals so I'm gonna say 0.3)
If A is the event that a person speaks a foreign language, and B represents the even that someone is under the poverty line, this quantity I just gave represents \(P(B|A)=0.3\)

Answer 55

Oh okay so say there is a 20% chance of an apple falling from a tree which has a 60% chance of apples growing on it..

So we write 20% probability based on 60% probability of it happening? 20|60?

Answer 56

no that is a different situation, because we aren't given that apples grow on this particular tree
a conditional probability would be "the probability of an apple falling given that it is on a tree", which in your case is about the same, since the only way for an apple to fall is for it to be on the tree, so that conditional probability is the same as the independent one...
but again, do *not* write the values of the probabilities where the symbols for what they represent go!
20|60 means "the probability of 20 given that we have 60"
again, I think we are talking about money or cards or something. use symbols for events and don't confuse them with their probabilies, which are numbers

Answer 57

your example is bad because it brings up the relationship between conditional probabilies and independence, which is a somewhat different topic that I'd rather not get sidetracked with.

Answer 58

Oh I apologize for that then!

I won't write the values directly :)

We'll use the language example instead!

Answer 59

no worrie :)
ok so do you understand the notation and setup of the problem as I stated it above?

Answer 60

Yeah!

50% of the people speak a foreign language and 30% of the people who speak a foreign language are below the poverty line

Answer 61

So we have 50% of all people speak a different language, and in that 50%, 30% of them live in poverty.

Answer 62

we know the conditional probability that a person is under the poverty line given they speak a foreign language is \[P(B|A)=0.3\]and the probabilty that any given person speaks a foreign language is \[P(A)=0.5\]we want to know what the probability that a person speaks a foreign language, given that we know they are under the poverty line, which is\[P(A|B)\]

Answer 63

we want to then use Baye's rule, which is\[P(A|B)={P(B|A)P(A)\over P(B)}\]we know \(P(B|A)\) and \(P(A)\), so now we need \(P(B)\)
do you know how to get that?

Answer 64

reminder: \(P(B)\) represents the probability that a random person is under the poverty line

Answer 65

No I don't get it how do this further, if you solve this for me, I'll do 2 - 3 more and show it to you right away!

Answer 66

I like your spirit :)
\(P(B)\) can be found with the total probability theorem.\[P(B)=P(A)P(B|A)+P(A^C)P(B|A^C)\]where \(A^C\) represents the "compliment" of \(A\), or "not A". In this case that is the event that a person does *not* speak a foreign language.
at this point I'd actually Like to change the probability of a person speaking a foreign language to 60%, i.e. \(P(A)=0.6\), because it will better illustrate the next part
is that ok with you?

Answer 67

Yeah I get it :) so if 30% speak a language and we call the A, A^c will be 70%, Am I correct? :)

Answer 68

yep :)

Answer 69

do yo understand that formula for the total probability theorem?

Answer 70

P(B) = Probability of A x Probability of B happening when A happened + Compliment of Probability of A x Probability of B happening when compliment of A happened.

Answer 71

yeah but i mean intuitively, in English, it translates to
"probability of B happening is the probability that B happened, given that A happened, plus the probability that B happened, given that A didn't happen "

Answer 72

this should make some sort of sense if you think about the statement carfully, it's similar tohow all the event of something hapenning + not happening is 1

Answer 73

Hahaha yeah that makes so much sense :D :D

Answer 74

great, so if P(A)=0.6 (60% speak a foreign language), and P(B|A)=0.3
what are P(A^C) and P(B|A^C) ?

Answer 75

P(A^C) = 0.4 and P(B|A^C) = 0.7

Answer 76

exactly :)
and *now* we can finally plug in the numbers

Answer 77

P(B)=0.6x0.3+0.4x0.7

Is this right?

Answer 78

yep

Answer 79

1.8 + 2.8 = 4.6

Answer 80

?

Answer 81

So P(B) = 4.6 we put the P(B) back in the baye's theorem again

Answer 82

how did you get those numbers?

Answer 83

0.6 x 0.3 + 0.4 x 0.7?

Answer 84

you are off by a decimal factor in your multiplication
you should NEVER get a number greater than 1 for any probability. If so, you know you messed up

Answer 85

1 means it definitely happened; i.e. has 100% probability of happening
P(B)=4 means a 400% probability
use your calculator

Answer 86

omg, I am so daft, I forgot. I can't believe myself sometimes..

Answer 87

it happens

Answer 88

0.18 + 0.28

Answer 89

sounds more reasonable

Answer 90

I hope I don't do these mistakes in the exam(a week away)

Answer 91

so now that we have P(B), we can find P(A|B), which is what?

Answer 92

0.39 because P(B|A)P(A)/P(B) = 0.3x0.6/0.46

Answer 93

correct, very good

Answer 94

I'll do those 3 other examples in 5min hang on :)

Answer 95

sure thing :)