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Why odds 167 and 333 in Pick 3 ?

Dr Pepper

Those are the odds for a box on a six way number and a double 167 and 333 but none explain how.  

Thoth's avatar - binary

A Boxed Nomatch number can be drawn 6 different ways.  If you played 123 boxed in could be drawn as 123, 132, 213, 231, 312 or 321 and you would win.  No matter what number you play there are ALWAYS 1000 possible outcomes to any Pick 3 drawing.  To get the odds for a boxed nomatch number simply divide the 1000 total possibilities by the 6 you need to win with!

1000 / 6 = 166.66... Rounds up to 167

A Boxed Double Digit number can be drawn 3 different ways.  If you played 223 boxed in could be drawn as 223, 322 or 232 and you would win.  No matter what number you play there are still ALWAYS 1000possible outcomes.  To get the odds for a boxed double digit number simply divide the 1000 total possibilities by the 3 you need to win with!

1000 / 3 = 333.33 Rounds down to 333


Dr Pepper

 Thank you for your help Thoth. Trying to understand the nuts and bolts of this is difficult for me. It must be for a lot of others too. Almost 100 views before someone came along that was able to help like yourself.

 I understood that there are 720 six way numbers in the 1000. There are also 270 three way doubles in the 1000.  

 From this very basic understanding I can get around and make the right bets .  In other words, it's twice as hard to hit a double as a single number. If you do hit a double they pay you almost twice as much. That's because there are twice the amount of singles as doubles to hit in the 1000. Doubles are harder.  At least I have that part down. 

 What I don't get is this.  If I put a dollar on 321 box single does that mean I average a win every 167 times? I just don't understand the part where it goes from 1 in a thousand to 1 in 167. Seems like it would go from 1 in 1000 straights to  1 in 120 boxes not 167. There are only 120 boxed six ways not 167 and thats where it gets hard for me. Those extra 47 just seem to come from no where. I know it's right of course but it just feels wrong. 

Thoth's avatar - binary

Your welcome.

I know the odds are a little confusing at first, especially when you look at the total possible boxed combinations versus the total straights. Think of it this way:

120 Nomatch Boxed combos each play 6 ways (120 x 6 = 720 Total straight)

90 DoubleDigit Boxed combos each play 3 ways (90 x 3 = 270 Total straight)

10 Triples only played one way each                  (10 x 1 = 10 Total straight)

120 + 90 + 10 = 220 Boxed (Note: You can't play Triples Boxed)

720 + 270 + 10 = 1000 Straight Combinations

If you were to play 321 boxed then any of the following 6 numbers could be drawn to make you a winner: 321, 312, 213, 231, 123, 132. 

If you were to look at a list of all the Nomatch numbers in boxed form- there would be 120 of them.  Those 120 would in turn represent 720 straight numbers on the list of 1000. Understand that there is no guarantee that a no match number is going to be drawn!.....So the odds would never be 1 in 120.  There is still 90 boxed doubles (270 straight ) and 10 triples that could be drawn instead. When figuring the odds you must account for all the total possibilities.


By the way,

If you were to play 321 boxed continuously you most likely wouldn't win an average of once every 167 games.  Unfortunately the lottery doesn't work like that.  Here's the harsh reality (mathematical fact!):  If you play the same nomatch boxed combination every game, you will have to play it for 116 consecutive games in order to have a 50% chance of winning on it!  After the first 116 games is over the next 116 games is a second 50% chance, and so on and so forth!  The consecutive 50% chances can never be added to equal 100%.  Its like flipping quarters and guessing heads or tails!  How many heads can you flip in a row???


Hope this helps and doesnt discourage you from from trying to win,


Dr Pepper

Pretend for a second that there were no doubles in the Pick 3 game. If a double comes we just keep spinning until only a Single number was achieved.  Now then. The odds would be 1 in 720 for a straight right? Now,would the odds at getting a Box be 1 in 120? 

 How would the odds change if we added back say 10 doubles only.Then another 10 doubles and another. Maybe that would help me see it or how that changes the odds.

 Thoth that 116 consecutive games to have a 50% chance etc. That's hard to bend into any kind of straight forward thought for me. Now I've hurt myself.        

Thoth's avatar - binary

Ok, supposed you had a gigantic lottery machine with 720 unique balls in it.  Each ball would have to have one of the 720 Nomatch numbers printed on it. That means that if u were to play 123 boxed one of the 6 balls with 123, 132, 213, 231, 312, or 321 would have to be drawn in order for you to win boxed.  Yes, the odds of you winning in a lottery arranged like this would be one in 1 in 120 (720 divided by 6 = 120).

Now if you threw 10 more balls into the machine (we will use all ten triples) then there would now be 730 balls in the machine.  Now if you played 123 boxed your odds would be 730/6 = 121.666.... (1 in 122 rounded).

Add 20 more balls (they could be doubles if you like) to the 730 (total=750) and your odds for getting 123 boxed would be 750/6 which equals 1 in 125.

You always have to divide the total possible outcomes by the amount of numbers that will make you a winner to get your odds.

Oh and by the way, if you look at the last 116 consecutive games in your state and mark off all the doubles and/or triples you should have right around 60 DIFFERENT NOMATCH BOXED numbers.  60 is 50% of the 120 possible.  If you try this little experiment remember that 789 is the same number as 897 or 879 etc... when listing boxed numbers.  So, if you see the same numbers (no mater the order) anywhere in the list they are considered repeats.

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