In its purest form deductive inference starts with various statement known to be true, carries out a logical sequence of steps using these statements, and arrives at one or more further statements. If the initial statements really are true and the reasoning steps are logically valid then the new statements will also be true. Further deductive inferences can then be made starting from these new statements and, repeating this many times, very long chains of deductive inference can be created.

Mathematics is just a collection of these inferential chains. Deep Mathematics is when the chains are long and 'good' Mathematics is when the resulting statements are deemed to be interesting (which is often a subjective judgement). This pure form of deductive inference does occur outside Mathematics but usually the inferential chains are very short; however, when it can be used, it has the edge over inductive inference in that the statements it arrives at will be believable beyond doubt.

However, in my previous post I promoted both deductive and inductive inferences as ways of making good judgements, or making good decisions as often as possible. The pure form of deductive inference described above seems to be an 'all or nothing' process (and, in that case, will usually be a 'nothing' process because if even a single one of the links in the inferential chain cannot be made then the final result will be worthless).

What rescues deductive inference from being largely useless outside Mathematics is the notion of probability which gives us a way of ascribing a likelihood of truth to the conclusion we have reached. This is not the place in which to write about probability but the sort of thing I mean can be appreciated by a simple example. If I know it is 90% likely (probability 0.9) that my roses have black spot fungus and my fungicide has a 60% success rate (probability 0.6) at treating black spot then, by applying the fungicide I have a 0.9 * 0.4 = 0.36 chance of my roses continuing to have black spot after applying the fungicide; therefore no black spot is a 1 - 0.36 = 0.64 chance.

If you didn't follow that, don't worry. The main message of this post is that such reasoning is not easy because our brains don't seem to have a good intuitive grasp of probability. Formal training in probabilistic assessment is almost essential in order to reliably calculate the odds that any particular event will occur. It alerts you to the common logical fallacies but is definitely not a guarantee you will not fall into one. The remainder of this post is a description of some of the counter-intuitive conclusions that you might come to in assessing the likelihood of an event.

#### The Birthday paradox.

At a party or 24 or more people it is more likely than not that there will be two people with the same birthday. Most people find this surprising. The justification is no more than a few lines of probabilistic calculation but you have to know what you are doing.#### The Monty Hall problem.

A game show contestant is asked to pick one of three doors. One of the doors conceals a valuable prize and there is nothing behind the other two doors. The contestant makes their choice and then the game show host (who knows which door conceals the prize) opens a door that was not picked to reveal no prize behind that door. The contestant is asked whether they wish to change their mind and go for the other unopened door. Should they change their mind or not?It is very tempting to believe that the contestant neither increases or decreases their chance of winning by changing and that the prize is behind either remaining door with a 50-50 chance. In fact they double their chance of winning by changing their choice. This so counter-intuitive that (see this Wikipedia article) almost 1000 PhD graduates, including the famous Paul ErdÃ¶s, were fooled.

#### Boy girl combinations

If someone says to you "I have two children and (at least) one is a boy" what is the probability that they have two boys? The logical trap is to reason that the remaining child is as likely to be a boy as a girl and therefore the answer is 50%. In fact the actual probability of two boys is 33% (probability 1/3). The reason that this problem fools people is that the 'sample space' (the set of equally likely different possibilities from which one has to choose) is not what, at first, you think it might be (indeed, this is often the pitfall that probability presents). The sample space is BB, BG, GB (these combinations being the order by age of the children). Only one out of three represents two boys.See how subtle this is? Had you been told "I have two children and the elder is a boy" then the 50% answer would have been correct.

But if you thought that was bad enough enough consider being told this. "I have two children. One is a boy born on a Tuesday". Now what is the probability of there being two boys? The answer is 13/27. This astonishing answer is discussed in a Science News article along with various caveats and is worth reading for reinforcing my warnings about the danger of trusting your probabilistic intuition.

If you are still with me but are beginning to feel that these apparent paradoxes are not important in the real world please think again. We have to make decisions every day and most of the time we act with imperfect knowledge. But that doesn't mean that we cannot improve the choices we make. We have to realise that some choices are more likely to be successful than others, and that there are ways to find these successful choices more often than not. The science of Probability is the key to such informed decision-making. Should we eat organic foods? Should we get a 'flu shot? How much Life Insurance should we buy? The list is long. And we should remember that our gut feel about probability is very likely to be unreliable.

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