Overlapping Patterns Change Probabilities

If we tossed a coin repeatedly and we looked for this pattern HTT

vs.

If we tossed a coin repeatedly and looked for this pattern HTH

We would intuit that the average number of tosses until seeing either of the patterns is the same.

It's not.

The actual average number of tosses until the first pattern occurs is 10.

The actual average number of tosses until the second pattern occurs is 8.

The explanation lies in the fact that the second pattern "HTH" overlaps itself.

One can achieve two instances of HTH in the pattern HTHTH. One cannot achieve two instances of HTT in any combination of 5 coin toss attempts.

True and False Positives and True and False Negatives Must be Counted in Percentage Analysis

Assume 100 persons of every 1,000,000 have HIV.

Assume a test for HIV is 99% accurate.

The result shows positive for a particular person.

What is the actual probability that the person who tested positive has HIV?

Intuitively, we want to say 99%.

It's not. It's not even close.

Of all those who test positive, less than 1 in 100 will actually have the disease!

Why?

There are two groups being tested, those who do have HIV and those who don't.

Of those who have HIV (100 people in a million), 99 people will test positive. One will FALSELY test negative.

Of the remaining people who do not have HIV (999,900 in a million), 989,901 (or 99%) will test Negative and 9,999 (1%) will FALSLY test positive.

So, of the million tested, 10,098 will test positive, but only 99 of those will actually have HIV. That works out to less than one percent chance, specifically, a 0.98% chance of having the disease if a person tests positive.

His final example was of an expert who testified in criminal trial.

Probability is Profoundly Affected by shared Causation

The trial was of a woman who had had two children die of SIDS (sudden infant death syndrome). The prosecutor was making a case for how unlikely two such deaths in the same home are.

The expert made the claim that the chance of one child with SIDS was about 1/8,500. Therefore the chance of two was therefore 1/8,500 x 1/8,500. This = 1/ 7,2250,000. Of course with the jury hearing from an expert that the chance of the events occurring by "chance" was so low... well the woman was convicted.

There are some major assumptions involved in this calculation. The largest being that what causes SIDS once is wholly independent of what causes SIDS twice. This is to say that NEITHER NATURE nor ENVIRONMENTAL FACTORS -both of which are common to both infants, are factors in SIDS. This is so unlikely as to be nearly impossible.

The woman's case was overturned. The expert is currently under investigation.