In my country bookmakers offer the opportunity to bet on the National Lottery. I am trying to work out the actual probability of certain events happening in order to compare with the odds offered by the bookmaker.

For example:

The lottery consists of 47 balls numbered 1 to 47 and six balls are drawn. The bookmaker allows the player to choose exactly three numbers and if and only if all of those three numbers are included in the six drawn by the lottery the bookmakers pays out at 650/1 for a probability of ~0.0015361. I know the bookie will have an edge factored into the price, I am just interested in how big an edge.

I'm having trouble in expressing the problem mathematically so any help would be greatly appreciated. Thanks.

The probability of hitting 3 numbers is:

C(44,3)/C(47,6) ~ 0.001233426

Fair odds are around 810/1.

In the formula above, C(n,k) is the number of different ways (combinations) you can pick k elements from a set of n elements. The denominator represents how many possible six-numbers draw you can have. The numerator are your favourable cases: in order to win, the draw must contain your three numbers plus any three from the remaining 44.

Thanks very much for the explanation. I had been trying different methods using the combination formula but it never occurred to me to express it as you did. The bookmaker certainly has a large edge on such bets!