Math calculation for probability of winning "Set For Life" (UK Lottery game)?

The probability of two independent events happening at the same time is the product of the two individual probabilities. 

A simple example is flipping heads on a coin (1/2) and rolling a 6 on a die (1/6).  The probability of both happening at the same time is (1/12).

So in your case, the probability of getting the Life Ball is 1/10.

Now you have to have the probability of getting 5 numbers picked out of 47.  There is 1 winner, so let's just focus on the number of possible outcomes for now.

The first number can be any of the 47.

The second number can be any of the remaining 46.

The third number can be any of the remaining 45, etc. until 5 balls are chosen.

47 * 46 * 45 * 44 * 43 = 184,072,680

But this includes duplicates (1 2 3 4 5 and 5 4 3 2 1 count as two different entries).   So to remove them, divide by (5! or 120)

184,072,680 / 120 = 1,533,939

So the probability of getting the 5 numbers is:

1/1533939

Multiply that by the 1/10 from the life ball to get:

1/15,339,390

Which matches what the web site states as the probability of getting the top prize (1 in 15.3m)