Derin from Woodhouse College gave a clear presentation of his reasoning:
 
At first draw, there are 6 balls in the bag, any one of your 3 numbers needs to be picked, this gives you a $\frac{3}{6}$ chance (or $\frac{1}{2}$).

For your second ball to be chosen, there are now five balls in the bag and there are two possible balls which if picked would lead to a win, therefore, your chance for the second ball is $\frac{2}{5}$. For the last ball, there are now four balls left in the bag, only one of which has your number painted on it, therefore, the chance of your ball being picked is $\frac{1}{4}$.

In order for all three of your balls to be picked sequentially, you must multiply the probabilities of each being chosen on their own, i.e:$\frac{3}{6}\times\frac{2}{5}\times\frac{1}{4}= \frac{1}{20}$ therefore you have a one-in-twenty (or 0.05) chance of winning.
 
Ayden from Melbourn Village College calculated the solution to the rest of the problem. He also noticed that the chances of picking two balls out of six and four balls out of six is the same.

If you had a 2 out of 6 ball lottery it would increase your chances of winning, which is $\frac{1}{15}$ as opposed to $\frac{1}{20}$.
Drawing 4 balls instead of 2 will keep the chance of winning the same, $\frac{1}{15}$.
Having a 1 ball lottery will dramatically increase your chance of winning, which is $\frac{1}{6}$.
Having a 5 ball lottery will give you the same chance of winning, $\frac{1}{6}$.
To have the least possibility of winning a10 ball lottery, you would need to pick 5 balls.
The probability of winning the national lottery $\frac{1}{13983816}$ this is often rounded to $\frac{1}{14000000}$.

Phil from Wilson's School provided more explanations and some shrewd insights on how to solve the last parts of the problem

If the Mathsland lottery is using a ten-ball lottery and wants to make the least chance of winning as possible, then you must need to pick 5 balls correctly to win. This is because if you look at the six-ball lottery the chance of winning was its lowest when three balls were needed to be picked correctly. This is half of three and either side of 3 balls, 2 and 4, the chance of winning went back up.

For example, when five out of ten must be picked, the chance of winning is $\frac{1}{2}\times\frac{4}{9}\times\frac{3}{8}\times\frac{2}{7}\times\frac{1}{6}= \frac{24}{6048}=\frac{1}{252}$, which is smaller than four balls ($\frac{2}{5}\times\frac{1}{3}\times\frac{1}{4}\times\frac{1}{7}=\frac{1}{210}$) and six balls, therefore if the Mathsland lottery is organising a ten-ball lottery 5 balls should be predicted as this results in the lowest winning chances.

The chances of winning the UK National lottery is $\frac{6}{49}\times\frac{5}{48}\times\frac{4}{47}\times\frac{3}{46}\times\frac{2}{45}\times\frac{1}{44}=\frac{1}{13,983,816}$ or approximately 1 in 14 million.

However you can also win money for guessing 3, 4 or 5 numbers and the chances are:
 3 numbers: 1/56.7
 4 numbers: 1/1032.4
 5 numbers: 1/55,491.3 recurring.
 
George, also from Wilson's School, compared the result obtained with the simulator to the theoretical result. Great work! Click here to see the file.

It is worth pointing out that picking 2 balls out of 6 balls is essentially the same as picking 4 balls out of 6 balls, since by picking 2 balls and dumping 4 balls, you are also choosing 4 balls to dump and keeping 2 balls. The two processes are equivalent. The same goes for picking 1 out of 6 and picking 5 out of 6, etc.

Well done to everyone!