Question:
In the January 11,1988,issue of the Oil&Gas Journal, R.A.Baker describes how the Bayesian approach can be used to revise probabilities that a prospect field will produce oil. In one case he describes, geological assessment indicates a 25% chance the field will produce oil. Further,there is an 80% chance that a particular well will strike oil given that oil is present on the prospect field.

1. Suppose that one well is drilled on the field and it comes up dry. What is the probability the prospect field will produce oil?
2. If two wells come up dry, what is the probability the field will produce oil?
3. The oil company would like to keep looking as long as the chances of finding oil are greater than 1%. How many dry wells must be drilled before the field will be abandoned
4. If the first well produces oil,what is the chance the field will produce oil?

Hi Wei,

Let event B={a field will produce oil} and event A_i = { i-th drilled well will produce oil}.
It is given that P(B)=0.25 and P(A|B)=0.8. Hence P(not A_i|B) = 1-0.8=0.2 for any i=1,2,... and P(not B) =1-P(B) =0.75. Moreover, it is obvious that P(not A_i| not B)=1 for any i=1,2,...

We made two assumptions to solve this problem. The first is that if oil is present in the field then the field is capable of producing oil. The second is that the probability that a particular well produces oil is independent of whether any previous well has produced oil, that is the events A_i are independent.

a)

We need to calculate
 
P(B| not A₁)= Bayes formula = P( not A₁|B)*P(B) / P(not A₁).
 
To calculate P(not A) we use the formula of total probabilities:

P(not A₁)
= P(not A₁|B)*P(B) + P(not A₁|not B)*P(not B)
= (0.2)*(0.25) + (1)(0.75)
= 0.8.

Substituting all numbers we obtain:
 
P(B| not A₁)= (0.2)*(0.25)/0.8 = 0.0625=6.25 %

b)

We need to calculate  
 
P(B| not A₁ and not A₂)= Bayes formula = P( not A₁ and not A₂| B)*P(B) / P(not A₁ and not A₂).
 
To calculate P( not A₁ and not A₂| B) we use independence:
 
P( not A₁ and not A₂| B)= P( not A₁| B)*P(  not A₂| B) =(0.2)*(0.2)=0.04.
 
To calculate P(not A₁ and not A₂) we use the formula of total probabilities:

P(not A₁ and not A₂)
=  P(not A₁ and not A₂|B)*P(B) +  P(not A₁ and not A₂| not B)*P(not B)
= (0.04)*(0.25) +(1)*(0.75)=0.76.

Substituting all numbers we obtain: 
 
 P(B| not A₁ and not A₂)=  (0.04)*(0.25) /(0.76)=0.0132 = 1.32%

c)

Now consider three wells and calculate  P(B| not A₁ and not A₂ and not A₃)
 
If this is less than 1% then three well will suffice. If not try four wells.

d)

Certainly, it will be 1 (common sense). Is that what you get when you apply Bayes Formula?

Andrei and Penny

Go to Math Central