The above video simply shows a dice rolling in slow motion.

This learning module considers the basics of such as can be seen when using one or more six-sided (or die, whichever you prefer as the singular). We also will take a look at the as well as what changes in terms of probabilities when events instead are . Note regardless that everything is easier given independence so, if at all possible, scientists tend to concentrate on independent events, at least to start with, only finding themselves dragged into non-independence when they absolutely have to and/or because a lack of independence appears to underlie an interesting phenomenon. We will consider in association with independent events what is known as the (which, unfortunately, happens to be a different thing from the or ). We also will consider what is known as the

We start with the .

The above video introduces the . Note the importance here of .

The following videos introduce us to the , which is a more challenging concept than the .

The above video introduces the , considering both mutually exclusive and not ; we’re interested particularly in the mutually exclusive, which is equivalent to the "when order does not matter" example.

The above video is a bit 'dry', but all examples are based on rolling dice.

The following video reiteration of the as applied to when order does not matter (odds of rolling, e.g., a 6). It also brings us back to doing biology.

Why do we care about ? Genetics for one, which this video considers.

The next two videos introduce graphing, which is a really important life skill completely 'independent' of whether you care anything about science. Key concepts are those of independent variable (usually the ) and dependent variable (usually the ) as well as an appreciation of how to fully utilize the "real estate" found on your graph.

The above video is also very dry, but makes its point reasonably well.

The above video describes how to graph. Very simply stuff, and (ironically) poor graphics, but makes all of the right points.

OK, totally corny, but also totally worth watching. ☺

The following video provides a brief introduction to the idea of .

This video describes how to calculate the associated with a data set.

Niall Ferguson in his book, the The Ascent of Money , suggests that we, as humans, have a lot of . Here the point is that therefore we do not tend to make for highly effective economic creatures but many of these points (or perhaps even all) also are relevant to why, without proper training and discipine, we also don't tend to make for highly effective . Indeed, the whole point of science, really, is to better understand the world through a striving towards reducing such biases in our own personal as well as collective perspectives as the following (pp. 345-346):

1. , which causes us to base decision on information that is more readily available in our memories, rather than the data we really need;
2. , which causes us to attach higher probabilities to events after they have happened () than we did before they happened ();
3. , which leads us to formulate general rules on the basis of insufficient information;
4. (or ), which means we tend to overestimate the probability that seven events of 90 per cent propbability will all occur, while underestimating that at least one of seven events of 10 per cent probability will occur;
5. , which inclines us to look for confirming evidence of an initial , rather than evidence that would disprove it;
6. , which inclines us to look for confirming evidence of an initial hypothesis, rather than falsifying evidence that would disprove it;
7. , whereby we allow irrelevant but information to influence our dicision;
8. , which prevents us from proportionately adjusting what we should be willing to sacrifice to avoid harms of different .
9. , which leads us to underestimate the within which our estimates will be robust (e.g. to conflate the 'best case' scenario with the 'most probable'); and
10. , which inclines us to abdicate individual responsibility when in an crowd.