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# Probability : Bayes Theorem and General Probability

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For a different medical application of Bayes' theorem, suppose one person in 1000 suffers an adverse reaction to a drug, and a simple test for this reaction is on offer. The test is said to be 95% reliable, meaning that if the person would suffer a reaction, a positive result comes up 95% of the time, and if they would not have a reaction, a negative result occurs 95% of the time. What can we conclude from the knowledge that Susie tests positive?
It is far too tempting to conclude that, as the test is 95% reliable, there is a 95% chance she would suffer a reaction. This answer is quite wrong. Let S = Susie tests positive, and let R = She would suffer an adverse reaction. We seek P(R|S), using R and RC as the partition for Bayes' theorem. The background information can be expressed as
P(S|R) = 0.95 and P(S|Rc) = 0.05,
while we also know P(R) = 1/1000. Hence
P(S)= P(S|R)P(R) + P(S|Rc)P(Rc) = 0.95 x 1/1000 + 0.05 x 999/1000 = 0.0509.
By Bayes' theorem, P(R|S) = P(S|R)P(R)/P(S) = 0.00095/0.0509 0.0187! When Susie tests positive, the chance she would suffer the reaction is under 2% ? the test is virtually useless, even though it can claim to be 95% reliable.

2.9 In the last medical example above, make one change in the parameters: it is n 1000, not 1 in 1000, who would suffer an adverse reaction. Compute the chance that Susie would suffer a reaction, given that she tests positive, as a function of n (1 &#61603; n &#61603; 1000).

2.10 One coin amongst n in a bag is double-headed, the rest are fair. Janet selects one of these coins at random and tosses it k times, recording Heads every time. What is the chance she selected the double-headed coin?

2.11 Assume that if a women carries the gene for haemophilia, any child has a 50% chance of inheriting that gene, and that is always clear whether or not a son has inherited the gene, but the status of a daughter is initially uncertain. Karen's maternal grandmother was a carrier, the status of her mother in unknown but Karen's sister Penny has one son, who is healthy.
(a) Find the chance Karen's first child inherits the haemophilia gene.
(b) Penny now has a second healthy son; repeat the calculation in (a)
(c) But Penny's third is a haemophiliac; again repeat (a).

2.12 The till in the pub contains 30 20 notes and 20 10 notes. There is a dispute about what denomination Derek used to pay his bill, and the initial assumption is that all 50 notes were equally likely. The barmaid, Gina claims he used a 10 note, Derek disagrees. Both are honest, but may make mistakes. Show that, using the information that Gina correctly identifies notes 95% of the time, the chance it was a 10 note is 38/41. Derek, who correctly identifies 20 notes 80% of the time, and correctly indentifies 10 notes 90% of the time, says he used a 20 note. Update your calculation.

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Note: (Before we begin solving Problems 2.9 - 2.12, let us briefly discuss the fundamental concept of Bayes' theorem.)

Bayes' Theorem
Assume that an event A can be partitioned into a set of sub-events Ai, where i = 1, 2, ..., n, such that and . Such (sub-)events Ai are called mutually exclusive. Note that A is a universal set if and only if . By definition, the conditional probability of event Ai, given event B can be computed as

or simply .
It can be seen that
.
In most situations, it is often that we know only the probabilities of the individual events and their a posteriori probability . Bayes' theorem allows us to find their a priori probability using the knowledge of all a posteriori probabilities . Consider the following property,

, where A is a universal set.
Therefore, we can find a priori probability from

This equation is known as Bayes' theorem.
In Problems 2.9 - 2.12, we will experience how Bayes' theorem can help find a priori probabilities.

For a different medical application of Bayes' theorem, suppose one person in 1000 suffers an adverse reaction to a drug, and a simple test for this reaction is on offer. The test is said to be 95% reliable, meaning that if the person would suffer a reaction, a positive result comes up 95% of the time, and if they would not have a reaction, a negative result occurs 95% of the time. What can we conclude from the knowledge that Susie tests positive?
It is far too tempting to conclude that, as the test is 95% reliable, there is a 95% chance she would suffer a reaction. This answer is quite wrong. Let S = Susie tests positive, and let R = She would suffer an adverse reaction. We seek , using R and as the partition for Bayes' theorem. The background information can be expressed as
and ,
while we also know . Hence
.
By Bayes' theorem, ! When Susie tests positive, the chance she would suffer the reaction is under 2% - the test is virtually useless, even though it can claim to be 95% reliable.

2.9 In the last medical example above, make one change in the parameters: it is n in 1000, not 1 in 1000, who would suffer an adverse reaction. Compute the chance that Susie would suffer a reaction, given that she tests positive, as a function of n where (1 ï‚£ n ï‚£ 1000).

Solution:
Denote
S = event that Susie tests positive
R = event that Susie would suffer an adverse reaction
According to the statement that "if the person would suffer a reaction, a positive result comes up 95% of the time", we know that the conditional probability that Susie tests positive given that she would suffer an adverse reaction is

On the other hand, the conditional probability that Susie tests positive ...

#### Solution Summary

Four problems involving Bayes' Theorem and general probability are solved. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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