My definition of "random" that I use here is "involving equal chances for each number".

I believe that:

a) This is not random, as clearly the number 2 has a far higher chance of coming up.

b) This is not a mistake - the chances are 1/(2^10) that this could happen, so it is unlikely but perfectly possible

c) This is not true - there is still a 1/2 chance of this happening, the coin is not conscious.

d) This is possible but not true - again, there is a chance (albeit quite small) that this could happen.

e) This is possible but we only have a small sample of the number so it could repeat after 16 decimals, for example.

f) This is true - if there is no pattern to the digits.

g) This is very nearly but not quite random - it is influenced by some environmental factors (an extreme example is growing up with a heavy weight on your head!).

h) i) This is not necessarily true - the rain might still happen but might cover the whole region.

ii) This is not true - there might be huge downpours all the time, that were unexpected, to generate the 30% chance.

iii) This is true assuming the weather behaves, and the weather forecaster has given out the same warning.

iv) This is a slightly odd way of making the statistic, but it could be used.

i) The weather is not totally random - for example, a drop in air pressure is linked to thunderstorms.

j) This is partially true - some remnants of the weather will affect the weather for tomorrow.

k) This is wrong - the balls are totally random. There is the same chance of winning withballs of one colour as winning with balls of varied colours.

l) This is wrong - it is almost impossible to choose any correct sequence. There is no more likelihood of the sequence being 1, 2, 3, 4, 5, 6 than 11, 24, 28, 34, 41, 45.

m) Chris is right - Anna is right using my definition, but Bob is right using a definition of "allowing equal chances of all digits".