Bayes’ Theorem Thomas Bayes was a British mathematician ( ).

Introduction  Bayes theorem is essentially an expression of conditional probabilities. More or less, conditional probabilities represent the probability of an event occurring given evidence. To better understand, Bayes Theorem can be derived from the joint probability of A and B (i.e. P(A,B)) as follows:

Bayes’ Rule: Derivation  Definition: Let A and B be two events with P(B)  0. The conditional probability of A given B is: The idea: if we are given that the event B occurred, the relevant sample space is reduced to B {P(B)=1 because we know B is true} and conditional probability becomes a probability measure on B.

Bayes’ Rule: Derivation can be re-arranged to: and, since also: The formula looks very complicated, but in fact it is easy to use if you remember that the denominator is the total probability of B.

Bayes’ Rule:  Why do we care??  Why is Bayes Rule useful??  It turns out that sometimes it is very useful to be able to “flip” conditional probabilities. That is, we may know the probability of A given B, but the probability of B given A may not be obvious. An example will help…

Three girls, Aileen, Barbara and Cathy, pack biscuits in a factory. From the batch allotted to them Aileen packs 55%, Barbara 30% and Cathy 15%. The probability that Aileen breaks some biscuits in a packet is 0.7 and the respective probabilities for Barbara and Cathy are 0.2 and 0.1. What is the probability that a packet with broken biscuits found by the checker was packed by Aileen? Example 1:

Solution1: Let A be the event ‘the packet was packed by Aileen’,B be the event ‘the packet was packet was packed by Barbara’,C be the event ‘the packet was packed by Cathy’,D be the event ‘the packet contains broken biscuits’. We are given P(A) = 0.55, P(B)= 0.3, P(C)=0.15 and P(D/A) =0.7, P(D/B) =0.2, P(D/C)= 0.1. We need to find P(A/D), so we use Bayes` theorem to ‘reverse the conditions’:

 Now P(D) is the ‘total’ probability of D, that is the probability that a packet contains broken biscuits. This can be found very easily from the tree diagram. The outcomes resulting in a packet with broken biscuits are shown with an asterisk.

packet

 P(D)= P(D/A) P(A)+P(D/B) P(B)+P(D/C) P(C) =(0.7)(0.55)+(0.2)(0.3)+(0.1)(0.15) =0.46  As shown in the tree diagram P(D/A) P(A)= (0.7)(0.55)  Therefore P(A/D)= (0.7)(0.55)/0.46 =0.837  The probability that a packet with broken biscuits was packed by Aileen is 0.837

Example 2: TThree children,Catherine,Michael and David, have equal plots in a circular patch of garden. The boundaries are marked out by pebbles. Catherine has 80 red and 20 white flowers in her patch, Michael has 30 red and 40 white flowers and David has 10 red and 60 white flowers. Their young sister, Mary, wants to pick a flower for her teacher. QQuestion (a): Find the probability that she picks a red flower if she chooses a flower at random from the garden,ignoring the boundaries.

Solution: (a)  if the boundaries are ignored : The possibility space S= (flowers in the garden) and n(S)= =240 Let R be the event ‘a red flower is chosen’,then n(R)= =120 P(R)= n(R)/ n(S) = 120/240= ½ The probability that Mary picks a red flower if she ignores the boundaries is 1/2

Question (b): Find the probability that she picks a red flower if she first chooses a plot at random. Solution (b): A plot is chosen first: Each of the three plots is equally likely to be chosen. Let C be the event ‘Catherine's plot is chosen’, then P(C) =1/3 With similar notation P(M) =1/3 and P(D) =1/3

The outcomes resulting in the event R are shown with an asterisk on the tree diagram

Now P(R) =P(R/C) P(C)+P(R/M) P(M)+P(R/D) P(D) =(80/100)(1/3)+(30/70)(1/3)+(10/70)(1/3) =16/35 The probability that Mary picks a red flower if she chooses a plot at random is 16/35 Note: the two different results for part (a) and part (b) are slightly surprising. In the first case, there is one group of flowers and each flower is equally likely to be chosen. In the second case, even though each plot is equally likely to be chosen,the proportions of red and white flowers within these plots are different.

Question (c): If she picks a red flower by the method described in (b), find the probability that it comes from Michael's plot.

Solution (c): Using Bayes` theorem: Now P(R/M) P(M) = (30/70) (1/3) = 1/7 (from tree diagram) And P(R) = 16/35 (from part (b) ) Therefore P(M/R) = 1/7/16/35 =5/16 Given that Mary picks a red flower, the probability that it came from Michael's plot is 5/16

Reference from : Website:   Book:  Fundamentals Methods of Mathematical Economics 4 th Edition (Page )

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