Probably good call too but I’m a tight bastard and don’t like paying for things.

]]>Good call. I’ve done a couple of their courses already. Will have a look. Cheers.

]]>I have the results of a sequence of 100 coin tosses which happens to consist of 50 heads and 50 tails.

I scan through this sequence to see if there is a run of six heads.

I find such a sequence.What is the chance of that the next item in the sequence is also a head?

You’re asking the wrong question. The right question is, if you scan through a *random* sequence of 100 coin tosses for a run of six heads, what is the *expected* probability that the next toss will be heads? Such a random sequence might happen to have exactly 50 heads, or not (probably not). It is the average probability of a head after a run of six heads, averaged over all sequences in the sample space that matters. This probability will be 1/2, assuming the coin is fair and the trials comprising each sequence are independent.

Check out Coursera. They offer numerous introductory on-line stats courses free of charge.

]]>My apologies: I failed to adequately qualify my statement. If we flip a coin, say, 4 times and obtain H H T H then the probability mass function of this result, its mean, and its standard deviation are all very different from the results we would obtain by flipping an unbiased coin a number of times that tends towards infinity. For any finite value of N (the number of trials/samples) the results may or may not be skewed from the theoretical value.

As I’ve stated previously, when N is small, say, 100 then a sequence of all heads would be an entirely valid outcome: because this sequence will occur many times in extremely long sequences. A short sequence of truly random numbers is simply a subset of an infinite sequence of truly random numbers; the starting point of the subset within the infinite set is entirely arbitrary.

Allow me to illustrate… Suppose I generate a dataset of truly random heads and tails coin flips and it contains 10^60 samples. It should not make any difference which particular subset of 100 consecutive samples I select from this dataset because it is, by definition, a truly random dataset. However, a sequence of 100 consecutive heads (or any other particular sequence that you can think of) *will* occur (probably many many times) in the whole dataset. To think otherwise would be committing the gambler’s fallacy.

I am in no way attempting to criticize your superb explanations. Your explanations are valid if, and only if, the actual N-sequence results obtained from a truly unbiased coin just happen to exactly match the theoretical results. The probability of this happening in practice reduces towards zero as the number of samples reduces towards 1.

As I stated previously, discrete random variables do not produce sequences per se. When we use them for the purpose of producing a sequence, then we attempt to analyse the sequence, we get confused each time we find a sequence that fails to match our expectations and/or our mathematical predictions. It isn’t that the sequence generator (the discrete random variable) is somehow faulty/biased, the fault lies entirely with us for our failure to understand that truly random events DO NOT produce sequences, they produce only an *unordered* set of results! The probability mass function, the mean, and the standard deviation are statistical parameters that are totally agnostic as to the order in which the samples arrive. Human cognitive biases prevent us from being likewise agnostic in our personal assessment of real world events and the datasets we look at.

Research psychologists have shown us that the order in which a set of facts/data is presented to us often has a huge impact on the way that we interpret, and react to, the set of facts. This is one of the many reasons why, I think, acquiring critical thinking skills is very important.

I sincerely thank everyone for taking the time to write their comments and for sharing their experiences, knowledge, humility, and humour.

]]>I have the results of a sequence of 100 coin tosses which happens to consist of 50 heads and 50 tails.

I scan through this sequence to see if there is a run of six heads.

I find such a sequence.

What is the chance of that the next item in the sequence is also a head?

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