'Probability' versus 'Possibility'
Probability in a lottery sense, is something that is strictly mathematical.
If you know how many lottery balls are going to be drawn from a specific quantity of differently numbered balls, you can use a formula to determine the probabilty that your choices will be drawn.
In the simplest of cases, you don't even need a formula. Suppose you have to pick a particular numbered ball (without cheating, of course) from a bag containing 49 differently numbered balls.
The probability is simply 1 out of 49.
Possibility. on the other hand, has little to do with Math, and everything to do with 'Yes, it can happen' and 'No, it cannot happen'.
Possibility is either 'Yes' or 'No, 'On' or 'Off', 'True' or 'False. simply, either it is possible or it isn't.
If before a lottery drawing, you choose 6 numbers for a future drawing, every single one of your choices is equally possible, and your six choices have exactly the same possibility of winning the big jackpot as anybody else's 6 choices.
The instant that first ball is drawn, the possibilty either remains 'True' because it matches a number you had chosen, or if that first ball isn't among your 6 choices, then the possibility of winning the 'big one' is dashed to 'False'.
And so it goes all the way up to the 6th drawn ball. The 'possibilty' remains 'True' as long as each drawn balls matches one of your 6 choices.
And, (it should go without saying), if the first 5 drawn balls have matched 5 of your 6 choices -AND- the 6th drawn ball matches, then 'possibilty' is no longer an issue.
It is now a 'certainty '!
Those Incredible Odds!
Does anybody really pay attention to how outrageously remote are the odds of winning a lottery jackpot?
Are the various state-run lotteries just playing the general population for a bunch of suckers?
The common consensus amongst the politicians & lottery directors is that high jackpots will cause more people to buy more tickets for each drawing.
They think the best way to get those jackpots up into the stratosphere (and to get more people to spend more money ), is to create a game with outrageous odds.
Seems like the strategy is working.
After perusing the list of lottery odds below, you might want to try our Lottery Odds Calculator . It will show complete odds for matching fewer than all the balls. and it even has a little Combinations and Permutations Calculator to show how greatly the results of the two calculations differ for picking lottery numbers.
Here's a rundown of the very worst odds to the not-quite-so-awful:
(none of 'em seem very good)
One chance in over Five Hundred Millions . (
- 622,614,630 = 6 of 90 = SuperEnalotto . without the Lucky Star! ( Italy )
One chance in over One Hundred Million. (
Good Grief!): ----------------------
- 258,890,850 = 5 of 75 + 1 of 15 ( ' Mega Millions ' started 19 Oct 2013 )
- 195,249,054 = 5 of 59 + 1 of 39 = Powerball (Jan 7th, 2009 - Jan 11th, 2012)
- 85,900,584 = 7 of 49 ( Canada 'Lotto Max ' ) (see ***Note below)
- 76,275,360 = 5 of 50 + 2 of 9 ( UK 'EuroMillions ' ended May 2011 )
- 76,275,360 = 5 of 50 + 1 of 36 ( 'Big Game' which became 'Mega Millions' 13 Jan 99 - 14 May 2002
- 62,891,499 = 7 of 47 ( Atlantic & Ontario 'Lotto Super 7 ' end 18 Sep 2009) (see ***Note below)
- 52,969,000 = 5 of 50 + 1 of 25 ( 'Big Game' which became 'Mega Millions' 06 Sep 96 - 12 Jan 99
- 47,784,352 = 5 of 44 + 1 of 44 ( The old 'Lotto Texas ' that ended 22 Apr 2006 )
- 45,057,474 = 6 of 59 ( New York 'Lotto ' )
- 41,416,353 = 5 of 47 + 1 of 27 ( California 'Super Lotto Plus ' )
- 29,144,841 = 5 of 47 + 1 of 19 ( 'Hot Lotto ' 19 states revised 12 May 2013 )
- 28,989,675 = 6 of 55 ( Philippine 'Grand Lotto ' )
- 25,827,165 = 6 of 54 ( Texas 'Lotto Texas ' begun 26 Apr 2006 )
- 24,435,180 = 5 of 45 + 1 of 20 ( South Africa 'Powerball ' )
- 22,957,480 = 6 of 53 ( Florida 'Lotto ' )
- 20,358,520 = 6 of 52 ( Illinois 'Lotto ' )
- 13,983,816 = 6 of 49 ( NJ, PA, WA, & WI in USA & Canada, UK, Hong Kong, 6aus49, etc. )
- 13,818,168 = 5 of 40 + 1 of 21 ( CT, MA, ME, NH, RI, & VT 'Win for Life ' )
- 12,271,512 = 6 of 48 ( Indiana 'Hoosier Lotto ' & Oregon 'Megabucks ' )
- 10,939,383 = 5 of 39 + 1 of 19 (old) 'Hot Lotto ' ended 11 May 2013
- 10,737,573 = 6 of 47 ( MI 'Classic Lotto ', GA & KY 'Decades of Dollars ' )
- 8,145,060 = 6 of 45 ( Ontario, CA & New South Wales, AU )
- 8,060,598 = 5 of 39 + 1 of 14 ( UK new 'Thunderball ' began 9 May 2010 )
- 7,059,052 = 6 of 44 ( Missouri 'Lotto ' & Arizona 'The Pick ')
- 5,245,786 = 6 of 42 ( 'Tri-State Megabucks ' ME, NH, VT )
- 6,096,454 = 6 of 43 ( Maryland 'Multi-Match Lotto ' )
- 5,034,400 = 5 of 32 + 1 of 25 ( Kansas 'Super Cash ' )
- 4,496,388 = 6 of 41 ( The old Arizona 'The Pick ' that ended March 18, 2007)
- 3,895,584 = 5 of 34 + 1 of 14 ( UK old 'Thunderball ' ended 8 May 2010 )
- 3,838,380 = 6 of 40 ( Louisiana 'Lotto ' )
- 3,262,624 = 6 of 39 ( DC 'Daily 6 ' & WI 'Super Cash! ', KY '3 Line Lotto ' )
- 2,986,522 = 4 of 44 + 1 of 22 ( Florida 'Mega Money ' )
- 2,760,681 = 6 of 38 ( Pennsylvania 'Win for Life ' )
- 2,718,576 = 5 of 31 + 1 of 16 ( 'Wild Card 2 ' ID, MT, ND, & SD )
- 2,598,960 = 5 of 52 ( Washington 'Quinto ' )
- 2,118,760 = 5 of 50 ( Indiana 'Mix & Match ' )
- 1,832,600 = 4 of 35 + 1 of 35 ( Texas 'Two Step ' & ME, NH, VT 'Weekly Grand ')
- 1,623,160 = 6 of 35 ( Delaware 'Multi-Win Lotto ' )
- 1,623,160 = 5 of 35 + 1 of 5 ( TN 'Tennessee Cash ' )
- 1,533,939 = 5 of 47 ( Minnesota 'Gopher 5 ' )
- 1,395,360 = 5 of 19 ( Pennsylvania 'Mix 'n Match ' permutation to match sequence Drawn)
- 1,353,275 = 4 of 77 ( Oregon 'Win for Life ' & Ontario & Atlantic 'Payday ' )
- 1,268,256 = 4 of 33 + 1 of 31 ( Kentucky 'Cash Ball ' )
- 1,221,759 = 5 of 45 ( 'Triple Play ' ME, NH, VT )
- 1,086,008 = 5 of 44 ( Missouri 'Show Me 5 Paydown ' )
- 962,598 = 5 of 43 ( Pennsylvania 'Cash 5 ' )
- 888,030 = 7 of 27 ( UK 'Daily Play ' )
- 850,668 = 5 of 42 ( Malta 'Super 5 Lotto ' )
- 749,398 = 5 of 41 ( Atlantic 'Bucko ' )
- 709,260 = odds of being struck by LIGHTNING
- 658,008 = 5 of 40 ( New Jersey 'Jersey Cash 5 ' )
- 649,740 = ROYAL FLUSH in Poker. any of the 4 suits
- 575,757 = 5 of 39 ( CA, GA, IL, MD, MI, MO, NY, OH, WA, W I )
- 501,942 = 5 of 38 ( Nebraska 'Pick 5' & South Carolina 'Palmetto Cash 5 ' )
- 435,897 = 5 of 37 ( LA 'Cash 5 ', MT 'Montana Cash ', & TX 'Cash 5 ' )
- 376,992 = 5 of 36 ( Florida 'Fantasy 5 ' & Indiana 'Lucky 5 ' )
- 324,632 = 5 of 35 ( AZ, CT, DC, IA, RI, SD )
- 278,256 = 5 of 34 ( New Mexico 'Roadrunner Cash ' & Virginia 'Cash 5 ' )
- 230,300 = 4 of 50 ( Louisiana 'Cash Quest '
- 201,376 = 5 of 32 ( Colorado 'Cash5 ' )
- 177,100 = 6 of 25 ( West Virginia 'Cash 25 ' )
- 169,911 = 5 of 31 ( Minnesota 'Northstar Cash ' & Wisconsin 'Badger 5 ' )
- 142,506 = 5 of 30 ( Pennsylvania 'Treasure Hunt ' )
- 105,625 = 2 of 26 + 2 of 26 ( '2 by 2 ' Kansas, Nebraska, & North Dakota )
- 100,000 = Pick 5 (Pennsylvania 'Quinto ' & Washington D.of C. 'Pick 5 ')
- 10,000 = 'Pick 4 ' (28 states + Wash. D.of C. )
- 1,000 = 'Pick 3 '
(33 states + Wash .D.of C. plus some Canadian provinces)
- 1 out of either 37 or 38 = Roulette Wheel (single or double zero)
- 1 out of 36 = Specific combination from a pair of dice
- 1 out of 6 = Roll of a single die
- An honest flip of a coin .
- Tomorrow's sunrise
- Life: Birth, learning, toil & troubles, taxes, happiness & sadness & everything inbetween, & (dangitall, ya can't take it with you), death.
*** Note: The Lottery Odds shown above are for matching the Drawn Ball Numbers with the picks on one board/panel on a playslip. Some lotteries, such as Canada's Lotto Max and Missouri's Lotto tout significantly lower odds because the player is forced to accept more than one board/panel/line/row for a minimum wager.
As with any lottery, the odds of winning the jackpot are reduced by dividing the odds for the jackpot by the quantity of boards/panels on a playslip. Using Super 7. a 7 from 47 game as an example, playing 3 boards reduces the odds from 62,891,499 to 20,963,833. In September of 2009, Super 7 was replaced by Lotto Max with even more terrible odds. The minimum 3 boards reduces the odds from 85,900,584 to 28,633,528 .
If you want more details about lottery odds & combinations (& permutations), & the odds of matching fewer numbers than the jackpot prize, use our Lottery Odds Calculator . You can also use it for calculating combinations & odds if you are investigating lottery wheels .
Where 'Common Sense' goes Awry
I think a person is using 'common sense' when they want to participate in a state-run lottery; and they want to see what numbers have come up in previous drawings.
Most state-run lotteries show the results of previous drawings, i.e. the winning numbers, on one of their website pages.
Some of the lottery web pages are now beginning to show the frequencies of the individual numbers' occurrences.
Here's a few examples:
What is utterly unamazing. is:
1) Every possible number eventually gets picked, and
2) The frequencies are all just about the same for any given set of numbers.
Think about that for a moment. kinda let those two important little factoids sink into your cranium.
Here's another factoid that is at least equally important:
Those little 'ping pong' type balls are as dumb as doorknobs! They don't have the ability to remember the last time they were drawn!
Equally important is the fact that they lack any sense of duty, i.e. they don't know that if a long time has passed since they were last drawn and/or that they are obligated to make an appearance now and then.
In other words, those little balls are totally devoid of conscience and consciousness.
Another factoid (no less important than the others, BTW) is that inbetween the official drawings that are usually televised, the lottery people continually run tests, i.e. the balls are weighed and checked in various ways, including to perform many untelevised drawings inbetween the official televised drawings.
And the last little factoid is that the sets of balls used in the official, televised drawings are seldom the same set for every drawing.
So what do all these facts tell you? And, more importantly, should they have any effect on your 'common sense'?
The facts tell you that the balls all have an equal chance at being drawn, whether the drawing is televised or not.
If that's so, then where does 'common sense ' go awry?
It's when a person starts thinking that some numbers are due to be drawn, or that some particular number or numbers are 'hot'. or when a person starts thinking that some mathematical formula based on past drawings is going to reveal one or more surefire numbers in the next drawing.
I used to think along those lines, that is, if I properly analyze the previous drawings, I'll be able to detect a pattern or some clue that some number is much more likely to be drawn (or has little chance of being drawn) in the next drawing.
Some folks like to analyze numbers, especially when they are organized into neat, finite sets like lists of previous lottery drawings.
Count me among the people in that group. I started writing analysis programs over twenty (20) years ago. My programs crunched those numbers in more ways than I can recall.
For all those years of effort, there's only one thing that I've been able to prove with a fair amount of certainty. That is, in any given drawing, any number has an equally random chance of being drawn.
Today you can buy (or however you chose to aquire them) software programs that do outstanding jobs of analyzing past drawings from every imaginable angle. The output from these programs will present you with lists, tables, and complex charts and graphs that are dazzling. But I've yet to see one that can reliably pick the numbers for the next drawing.
And this is where 'common sense ' goes awry.
Common sense: wrong = Past drawings offer some clue that one number is more likely (or less likely) to be drawn than another.
Common sense: right = If anyone ever devises/scripts a program that reliably predicts the future, then the lotteries as we know them today, will cease to exist. Period!
Have you ever picked the wrong numbers?
Guess what? Unless you chose your numbers after the drawing, you've never picked the 'wrong' numbers.
Since the drawing happens after you've made your picks, then, technically speaking, any number drawn that doesn't match one of yours, is wrong. (It might match somebody else's, but it's wrong as far as you're concerned.)
That is to say, only the drawn numbers can be right or wrong, not yours.
Either the numbers drawn correctly match yours, or if they don't, then the drawing presented the 'wrong' numbers.
Admittedly, it's small consolation when the drawn numbers don't match your choices, but you are not at fault. you didn't pick the 'wrong' numbers, the lottery did.
Combinations versus Permutations
First off, just be glad that most lotteries pay off on 'Combinations' of numbers and not on 'Permutations' of numbers.
The mathematical term 'Combination' can be confusing because we like to think about how many different ways we can 'combine' a group of things.
Say we had an apple, an orange, and a pear. We use the word 'combine' when we should be using the word 'arrange'. We can arrange the set as an orange on the left, an apple in the middle, and a pear on the right; or maybe put a pear on the left, an apple in the middle, and an orange on the right.
There are 6 possible ways to arrange the fruit from left-to-right. Each of those 'arrangements' is a permutation, not a mathematical combination.
The group of 3 fruits, without any concern for arrangement, is a 'combination' of 3 fruits.
A mathematical combination can be a subset of a set. Of the 3 fruits, a person could ask, "How many combinations are there of just 2 fruits from the set of 3?"
One combination is an apple and an orange, another is a pear and an orange, another is an apple and a pear. Order is unimportant, i.e. apple and orange is the same as orange and apple. each grouping is still just one mathematical combination.
There are 3 possible combinations (subsets) of fruits taken 2 at a time from a set of 3.
Reiterating, there are 6 permutations (arrangements) and only 3 mathematical combinations (subsets).
If the sequence is important, then it's a Permutation. If sequence is not important, it's a Combination.
Let's look at a lottery example, say a game where 5 balls (a subset) are drawn from a set of 39 balls:
In a 5 ball lottery drawing, say the numbered balls are drawn in the sequence of #30, #24, #12, #39, and lastly, #22.
That sequence is just one of 125 possible arrangements/sequences (permutations) of those drawn balls. They could have come down the chute as #22, #24, # 30, # 39, and lastly, #12, but however they are sequenced, there is only one subset(combination) that includes those 5 balls.
That subset is just one of the 'combinations', of which there are 575,757 possible combinations from the set of 39 balls.
How many different ways can the subsets of 5 balls from a set of 39 balls be arranged (permutated)? The answer is 69,090,840 .
Here's a link to our Lottery Odds Calculator . See for yourself what a huge difference there is between a mathematical combination and a permutation.
Pick 3 and Pick 4 lottery games are some of the very few games where you can bet on either one of the permutations or the combination.
In Pick 3, betting on the sequence of the 3 drawn balls is called a 'Straight', that is to say, you're betting on 1 of 6 possible permutations (if all 3 balls have a unique number). If you want to bet on all possible permutations (arrangements) of the balls, you bet on the 'Combination', which is termed a 'Box'.
Again, just be glad that lotteries pay off on 'Combinations' of numbers and not on 'Permutations' of numbers. The odds in a 6 balls from 49 balls game would go from millions. 13,983,816 combinations to billions. 10,068,347,520 permutations!
One of the State of Pennsylvania's Lottery games is called Mix 'n Match. It might sem like relatively easy 'pickins' in that you only have to match 5 ball numbers out of 19.
As Combinations go, the odds should be one in 11,628 to get the top prize. That would be just a tad more difficult than matching a Pick 4 game where the odds are one in 10,000 .
But wait just a dad-gummed minute! In order to win, you also have to match the sequence in which the balls were drawn. That is to say, if the Balls were drawn as 05, 04, 03, 02, 01 & you picked 01, 02, 03, 04, 05, then your only match would be on the third ball drawn (03). The odds of top prize in this game is a Permutation of one in 1,395,360 !
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