### The Lady or the Lions

The King showed the Princess a map of the maze and the Princess was allowed to decide which room she would wait in. She was not allowed to send a copy to her lover who would have to guess which path to follow. Which room should she wait in to give her lover the greatest chance of finding her?

Four fair dice are marked differently on their six faces. Choose first ANY one of them. I can always choose another that will give me a better chance of winning. Investigate.

### Nines and Tens

Explain why it is that when you throw two dice you are more likely to get a score of 9 than of 10. What about the case of 3 dice? Is a score of 9 more likely then a score of 10 with 3 dice?

# Misunderstanding Randomness

##### Stage: 3 Challenge Level:

Patrick from Woodbridge School sent us the following comments:

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".