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How to Estimate Drawing Chance – Transcript

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How to Cypher Drawing Chance

Updated on December 13, 2016



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Dez has been a mathematician since score schooltime and has a professional’s point in Applied Math.

As a mathematician, I sustain ne’er purchased a drawing slate. I discovery the odds cheerless and bear ne’er had fortune in victorious anything from these kinds of games.

This hub is all astir calculative drawing chance or odds. In edict to pass more relevant to me, I distinct to pedestal it on the Grandlotto 6/55, the drawing punt with the biggest pillage money hither in the Philippines. Thither testament be two dissimilar cases discussed in the hub: the chance of victorious the gamey with all six numbers twin, and the chance of having n numbers duplicate.

Rules of the Drawing Biz

It is incessantly crucial to determine the rules of any back earlier active in it. For the Grandlotto 6/55, in decree to win the pot loot, you bear to equal six numbers from a kitty of 55 numbers ranging from 1-55. The initial payout is a minimal of P20 (or about $0.47). It is likewise potential to win around money if you are able-bodied to equal iii, foursome, or pentad numbers of the victorious compounding. Bill that the fiat of the fetching combining hither does not count.

Hither is a tabularise for the prizes you can prevail:

No. edubir reviews of Twinned Nos.

Plunder Money (in Php)

Loot Money (in $)


minimal of 30 billion











Roughly Chance Concepts

Earlier we first with the calculations, I would alike to discuss Permutations and Combinations . This is one of the staple concepts you hear in Chance Possibility. The briny dispute existence that permutations reckon decree to be significant, patch in combinations, edict isn’t significant.

In a drawing fine, transposition should be ill-used if the numbers in your tag suffer to mate the ordering of the haul for the victorious thread of numbers. In the Grandlotto 6/55, gild is not significant because arrivederci as you birth the fetching set of numbers, you can win the plunder.

The adjacent formulas just give for numbers without repeat . This way that if the numeral x is worn, it cannot be worn again. If the turn careworn from the set is returned ahead the succeeding haul, so that has repeating.

This is the rule for Permutations, where fiat is crucial. | Root

This is the recipe for Combinations, where gild is not crucial. | Germ

, where n! = n * (n – 1) * (n – 2) * … * 3 * 2 * 1.

Line that based on the formulas tending, C(n,k) is invariably less than or capable P(n,k). You bequeath see afterward why it is significant to shuffle this eminence for conniving drawing odds or chance.

How to Estimate Drawing Chance for 6 Coordinated Numbers

So now that we recognise the staple concepts of permutations and combinations, let us recuperate to the instance of Grandlotto 6/55. For the back, n = 55, the summate issue of potential choices. k = 6, the numeral of choices we can piddle. Because edict is not crucial, we bequeath use the recipe for compounding:


These are the odds or the tally numeral of potential combinations for any 6-digit turn to win the plot. To obtain the chance, barely part 1 by the bit supra, and you bequeath get: 0.0000000344 or 0.00000344%. See what I think by uncheerful odds?

So what if we’re talk almost a unlike drawing plot where ordering does issue. We bequeath now use the replacement recipe to get the next:


Comparison these two results and you volition see that the odds for acquiring the victorious combining where ordering matters has 3 extra nil’s! It’s sledding from most 28 1000000:1 odds to 20 trillion:1 odds! The chance of taking for this lawsuit is 1 shared by the odds which equals to 0.0000000000479 or 0.00000000479%.

As you can see, because the replacement is ever greater than or capable the combining, the chance of victorious a gage where club matters is invariably less than or capable the chance of fetching a back where edict does not subject. Because the jeopardy is greater for games where rescript is mandatory, this implies that the payoff moldiness too be higher.

How to Estimate Drawing Chance with Less Than 6 Duplicate Numbers

Because you can too win prizes if you sustain less than 6 duplicate numbers, this department bequeath establish you how to estimate the chance if thither are x matches to the fetching set of numbers.

Beginning, we indigence to breakthrough the figure of way to prefer x victorious numbers from the set and manifold it by the act of slipway to prefer the losing numbers for the left 6-x numbers. Believe the turn of slipway to opt x fetching numbers. Because thither are solitary 6 potential fetching numbers, in gist, we are alone choosing x from a kitty of 6. So, because gild does not subject, we get C(6, x).

Succeeding, we see the routine of shipway to select the unexpended 6-x balls from the syndicate of losing numbers. Because 6 are taking numbers, we deliver 55 – 6 = 49 balls to prefer the losing numbers from. So, the issue of possibilities for choosing a losing egg can be obtained from C(49, 6 – x). Again, ordering does not count hither.

So, in rescript to figure the chance of victorious with x twin numbers out of a potential 6, we demand to fraction the event from the late two paragraphs by the aggregate act of possibilities to win with all 6 twin numbers. We get:


If we compose this in a more cosmopolitan manikin, we get:


, where n = add numeral of balls in the set, k = summate turn of balls in the fetching compounding for the kitty swag, and x = amount numbers of balls twinned the taking set of numbers.

If we use this recipe to compute the chance (and the odds) of taking the Grandlotto 6/55 with alone x coordinated numbers, we get the followers:

x matches



Odds (1/Chance)


C(6,0) * C(49,6)/C(55,6)




C(6,1) * C(49,5)/C(55,6)




C(6,2) * C(49,4)/C(55,6)




C(6,3) * C(49,3)/C(55,6)




C(6,4) * C(49,2)/C(55,6)




C(6,5) * C(49,1)/C(55,6)




C(6,6) * C(49,0)/C(55,6)



How to Select the Fetching Numbers in Drawing

As you can see from the mathematics therein hub, the chance of taking the drawing is the like for any 6-number combining usable in the Grandlotto 6/55 plot. This is likewise applicable for over-the-counter drawing games out thither.

As I was researching for this hub, I came crosswise links that aforesaid ne’er prefer numbers that are serial, care from 1-6 or about such gimcrack. Thither is no such mysterious to taking the drawing! Apiece figure is as evenly probable to scrape in the hooking as the following issue.

If you are uncoerced to look the really niggling chance of taking the drawing, I say go take any bit you deprivation. You can groundwork it on your birthdays, particular years, anniversaries, favorable numbers, etcetera. Good commend that with gravid endangerment comes gravid honor!

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