Calculating lottery OVERALL odds?

Question

I've seen various formula examples of how to calculate odds for a given number of picks, but what about the overall odds? For example Washington State's game, Hit 5. According to their web site, the overall odds are 1: 8.8. How is this calculated?

Number of balls: 39

5 of 5 1: 575,757

4 of 5 1: 3,387

3 of 5 1: 103

2 of 5 1: 9.6

Overall Odds 1: 8.8

Answer 1

The simplest formula is picking all 5 correct out of 39. There is only one correct combination. The total number of possible combinations is given by:

39C5 = (39!)/(34!*5!) = 575,757

So, you get 1 out of 575,757

For the other numbers, you need to compute how many ways you can get exactly that many correct numbers.

This is a two-part formula: First compute the number of combinations of the exact number of correct numbers from the set of winning numbers; then compute the number of combinations to get the exact number of incorrect numbers for the set of non-winning numbers. Then multiply the two.

To state it in terms of the odds, divide the answer into the total number of possible tickets.

Number of ways to match 4 numbers out of the 5 winning numbers:

5C4 = 5!/(4!*1!) = 5.

Number of ways to get 1 losing number out of the 34 losing numbers:

34C1 = 34.

Total number of ways to get exactly 4 numbers  = 34 * 5 = 170.

Odds of exactly 4 correct = 170/575757 = 1 in 3386.805, rounded to 1 in 3387.

For the others:

3 of 5: 5C3 * 34C2 = (5!/(3!*2!)) * (34!/(32!*2!)) = 5,610, or 1 in 102.63

2 of 5: 5C2 * 34C3 = (5!/(2!*3!)) * (34!/(31!*3!)) = 59,840, or 1 in 9.62

To get the overall odds, add up the total number of winners, and divide into the total possible tickets:

1 + 170 + 5610 + 59840  = 65,621 or 1 in 8.77

Answer 2

1 / ((1 / 575,757) + (1 / 3,387) + (1 / 103) + (1 / 9.6)) = 8.8