Conditional Probability

by Gerald L Fitton

Introduction

The Outcome of many Events is uncertain and so we resort to estimating the probability that the Statistic will take particular values. What many of you who are not professional statisticians might not realise is that this probability does not have an absolute value but that the probability of any event depends upon the extent of your knowledge.

In this article I shall develop that theme and give a couple of concrete examples.

Imperfect Information

For my first example let's consider a pregnant (soon to be) mother. On the basis that about half the children born are male we could say that the probability of a baby being a boy is 50% (actually it's nearer 53%). But is this really true?

Allow me to complicate this example. Mum-to-be has a test. A side result of this test is known to the Gynaecologist but Mum doesn't know nor does she want to know.
The Gynaecologist now reckons that he can predict the sex of the baby with 90% accuracy whereas Mum knows that the probability of a boy is still only 53%!

Let's take one more step in our imagination. Suppose I was all knowing (rather like God), I would have a 100% probability of predicting the sex of the baby - after all the baby's sex is a definite fact (even if we mortals don't know what it is).

So who is right about the probabilities?

The answer is: "All are right!" There is no such thing as an absolute value for a probability. The value you give to a probability depends on the extent of your knowledge.

The Value of Imperfect Information

The following is part of an assignment which I set my Decision Making students when I taught at the local College. I recommend using a spreadsheet to calculate the answer but you can do it with a sort of decision tree if you don't like spreadsheets. I include it as part of this article (but without solution) so that those of you willing to study the concept of relative (rather than absolute) probabilities will come to understand that even imperfect knowledge is financially valuable and this is because it does change probabilities (but not the absolute facts).

There is a concept used by those in this branch of mathematics which (without wishing to sound blasphemous) is called The value of a 'Hot Line to God'. Less blasphemously it is expressed as The Value of Perfect Information. When reading through the example below remember that God does know, right from the beginning, whether there is oil in the property under consideration (and the sex of the unborn baby). Us lowly mortals can never be sure of anything (unless we have Faith - but that's another story) in the way God can!

The Problem

The Western Standard Petroleum Company (WSPC) is a newly established small oil company. At present it is considering purchasing a plot of land where its preliminary survey indicates that there is a 60% chance of finding oil worth £1M. It has an option to buy the land for £0.3M and drill at a cost of £0.2M.

As an alternative WSPC could commission a survey from the well established and reputable Eastern Oil Survey Company (EOSC) without first buying the land. If the survey indicates that oil is present then WSPC will buy the land for £0.3M and drill at a cost of £0.2M (spending £0.5M).

Based on experience, such surveys by EOSC are correct 90% of the time. Calculate the maximum value to WSPC of an EOSC survey which is correct 90% of the time.

Suppose that EOSC offer to do the survey for £0.1M. On the basis of Expected Value alone should WSPC ask EOSC to conduct the survey or not?

Assume now that WSPC are willing to accept the loss of the cost of the survey, £0.1M, if the survey indicates that oil is not present. However, WSPC are not willing to accept liability for the cost of buying the land and drilling, £0.5M, if the EOSC survey indicates incorrectly that there is oil. WSPC asks EOSC to indemnify them against an incorrect indication that there is oil present when there isn‘t. EOSC agree to take this risk in house at an extra charge. What is the minimum extra charge which EOSC should make for indemnifying WSPC against their extra loss (over and above the £0.1 for the survey) if no oil is found? Work this out on the basis that EOSC do not wish to do other than break even in the long term on their in house insurance business.

EOSC decide to charge this minimum amount. On the basis of Expected Values alone should WSPC commission the survey and pay the extra charge?

Does this insurance arrangement change the Expected Value of the project to WSPC? If not then what has changed?

Gerald's First Law of Gambling

As a by-the-way (and in conclusion) may I quote Gerald's First Law of Gambling:

Never bet on certainties

I have an interesting anecdote underlining this Law which dates back to the time when my wise old Physics Master insisted that I bet a tanner (six old pence) against him on the outcome of an experiment that I was certain of (I'd seen the experiment before). When he won (the conditions were slightly different but I didn't know that) he refused to take my tanner on the grounds that he "Never took his winnings when he was betting on a certainty".


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