Mathematics of the Lottery in Hong Kong - Mark Six // 六合彩的數學
Introduction 介紹
Mark Six is the only legal lottery in Hong Kong. In this game, 7 balls will be drawn out of 49 balls with 1 to 49 marked on them. The first 6 numbers drawn are called "drawn numbers", while the last one drawn is known as the "extra number".
六合彩為香港唯一合法彩票。六合彩中共有49個球，每個分別印上1至49。7個球會被攪出，當中首6個被稱為「攪出號碼」，而最後一個則被稱為「特別號碼」。
Prizes 派彩
Each lottery ticket costs HKD 10. The prize fund is formed by the total amount of tickets sold after the deduction of taxes, commission and charity purpose.
每張彩票售價為HKD 10。彩池由所有售出的彩票扣除稅額及慈善用途而組成。
Prizes for 4th division to 7th division are fixed amounts, while the prizes of 1st division to 3rd division would be a percentage of the prize fund after deduction of 4th division to 7th division prizes. For 1st division to 3rd division, the amounts would be split equally if there are more than 1 winner.
四獎至七獎為固定銀碼，而頭獎至三獎為彩池扣除四獎至七獎的派彩後的某個百分比。頭獎至三獎若有多於一張彩票中獎，則由所有中獎彩票均分。
The following is a table of prizes in Mark Six:
以下是六合彩的中獎方法:
| Prize | Criteria | Amount (HKD) |
|---|---|---|
| 1st Division | All 6 drawn numbers | 45% of prize pool after deduction |
| 2nd Division | 5 out of 6 drawn numbers + extra number | 15% of prize pool after deduction |
| 3rd Division | 5 out of 6 drawn numbers | 40% of prize pool after deduction |
| 4th Division | 4 out of 6 drawn numbers + extra number | 9600 |
| 5th Division | 4 out of 6 drawn numbers | 640 |
| 6th Division | 3 out of 6 drawn numbers + extra number | 320 |
| 7th Division | 3 out of 6 drawn numbers | 40 |
Very often the amount of 1st division prize can reach 10 million HKD, that is about 1.3 million USD!
很多時候頭獎的金額可以高達一千萬港元!
Probability of winning 得獎機會
1st Division 頭獎
There is a total of 49C6 possible combinations of numbers that you can bet (Remember you can mark only 6 numbers out of 49 numbers). Out of all these combinations, there is only one way to win. Therefore the probability is 1 / 49C6 = 1 / 13,983,816.
你總共有49C6種不同方式去填劃彩票。其中只有一個得獎方法，所以概率為1 / 49C6 = 1 / 13,983,816。
2nd Division 二獎
Again the denominator is 49C6. To form a ticket that can win the 2nd division prize, you need to include any 5 numbers out of the 6 drawn numbers, which gives 6C5. The remaining number on your ticket must be the extra number in order to win. Therefore, the probability is 6C5 / 49C6 = 1 / 2,330,636.
分母同為49C6。要產生一張中二獎的彩票，你需要在6個攪出號碼中包含5個，亦即6C5。你的彩票的另外一個號碼必需為特別號碼。因此概率為6C5 / 49C6 = 1 / 2,330,636。
3rd Division 三獎
Again the denominator is 49C6. To form a ticket that can win the 3rd division prize, you need to include any 5 numbers out of the 6 drawn numbers, which gives 6C5. The remaining number on your ticket can be any number that is not drawn, so there are 42 numbers for you to choose from. Therefore, the probability is 6C5 * 42 / 49C6 = 1 / 55,491.33......
分母同為49C6。要產生一張中三獎的彩票，你需要在6個攪出號碼中包含5個，亦即6C5。你的彩票的另外一個號碼可以為任意一個不被攪出的號碼，因此有42個可能性。因此概率為6C5 * 42 / 49C6 = 1 / 55,491.33......。
Can you try to calculate the probability of winning the 7th division? The first one to give the correct answer will receive 1 SBD! (Show your steps)
你能計算出七獎的得獎機會嗎? 首先計算出正確答案者將可獲得1 SBD! (請顯示步驟)
Snowball Draws 金多寶
A certain percentage of the prize fund is set aside for the Snowball Draws. This Snowball Draw will be included in the 1st division prize occasionally to make it more attractive. For example, tonight there will be a Snowball Draw of HKD 80 million!
獎池的某個固定百分比會被放入金多寶。金多寶在某些六合彩中會被加入至頭獎獎金，例如今晚的六合彩就會有八千萬的金多寶獎金!
That's why I have purchased a lottery ticket today as well! XD Please wish me for the best of luck!
因此我今天就購買了一張彩票，希望我可以中獎吧! XD
6C3 * 42C3 / 49C6 = 1/60.905
Excellent! :) I will distribute the 1 SBD to you when I have spare money haha (i am not lying you can check my wallet, it's empty lol)
Not need to rush la :)
Just sent 1SBD to you, thanks for participating :)
帮你点D啦。:-)
六合彩我都帮称过。
謝謝啦 :-)
有中過六合彩嗎?
没有。住两个月买过好几期。捐钱给马会。
幸好馬會有部分投注款額是慈善用途，買六合彩的錢就當是做善事吧
算是吧。
七奬(3个) =1/49C6 X 246820
1.76%
謝謝參與，可是差一點點啦
我一直以為是50%/50%, 一係中一係唔中 (JK)
(8000萬唔記得買!!!!)
你還有時間~~
no la, i dun have jc account T-T
unless no one win~
冇錯! 買一張就50%中，兩張就50% + 50% = 100%，中硬 :D
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Hi again, can you please add some links to your recent articles on the Six of the Best contest page. I would like to encourage some discussion as there are very few people voting. Thanks and good luck!