Calculating Lottery Odds (Mexican game "Gano Gato"; playpiece looks like tic-tac-toe)

[mika1]

Calculating Lottery Odds (Mexican game "Gano Gato"; playpiece looks like tic-tac-toe)

Calculating lottery odds is relatively straight forwards, but this one has got me very confused which im trying to figure out

It’s a Mexican lottery game (Gana Gato) where you get prizes for matching correct numbers in lines on what looks like a tic, tac, toe box (see attached image)

.

You choose a number for all the outside boxes between 1 and 5, then the middle box is a joker where it can be any number and be different for each line (so you can have 1-2-3 along the middle and then 2-3-4 across a diagonal)

Im trying to figure out what the odds are for matching 1 line all the way up to all 8 lines. 1 Line is simple, as its the odds of matching a line using the joker (4/25) + the odds of matching one line using those without a joker (4/125) which = 0.192 = 1 in 5.2. I continued this methodology all the way to matching 8 lines which came out as 1 in 1,383,195,135.41 which just cannot be right, otherwise there would pretty much never be a winner


Can anyone help me out working all the way up from matching 2 lines to matching 8 lines? Would be extremely helpful

 

[tkhunny]

1) To what are you trying to match? Is there a drawing?
2) How did you get 25 and 125? Are you SURE it's that simple?
3)How did you extend your method to 2 lines?

 

[mmm4444bot]

1) To what are you trying to match? Is there a drawing?

Yes, eight numbers are drawn (each being 1, 2, 3, 4, or 5), to fill the outer eight squares in the grid. You complete a ticket beforehand, picking your own numbers, hoping to match as many "lines" as possible. A line is three numbers in any row, column, or diagonal. The center square is wild, and you may choose its value differently for each line.

For calculating the odds on winning one line, I think mika1 first reasoned that there's a 1-in-5 chance of picking the right number for each square.

For each line including the wild square, you would only need to get two squares correct:

1/5 · 1/5 = 1/25

Four such lines are possible, so:

1/25 + 1/25 + 1/25 + 1/25 = 4/25

For each line not including the wild square, you would need to get three squares correct:

1/5 · 1/5 · 1/5 = 1/125

Four such lines are possible, so:

1/125 + 1/125 + 1/125 + 1/125 = 4/125

That's my guess for how mika1 arrived at 4/25 and 4/125.

I'm not sure that mika1 considered things like:

If you match seven lines, you automatically get 8 lines, and you win first place (the jackpot).

If you match either rows 1&3 or columns 1&3, then you automatically match both diagonals, and you win fourth place.

Anyway, googling gives information; one Mexican web site states the odds of winning the jackpot as 1 in 390,625. Another web site states the odds of winning at least one line as 1 in 6. :cool: