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# Lotto Probabilities

Page with 7+2 calculations (before 7+3 calculated here)

Earlier the Norwegian Lotto only draws two extra numbers. After they changed the rule and started draw tree numbers the probabilities changed as follow:
```With three(new) instead of two(old) extra drawn numbers, the probability
for winning
* 7 same
* 6+1 increase with 50% (14 -> 21 )
* 6+0 decrease with 4% (175 -> 168 )
* 5 same
* 4+1 increase with 44 % (21875 -> 31535)

This means
* There are 9960(31535-21875) more ways to win out of 5379616 possible ways
* There are  2.82% bigger chance of winning (966000/29436 = 32.82)
* But still you will only have a 0.73% (3909600/5379616) chance of winning,
up from previously 0.55% (2943600/5379616)
(ie you will win 1 out of every 138 time)
```

The probability of winning in the Norwegian lotto is as follow
 Probability (%) 7 0.0000186 6+1 0.00039 6 0.0031 5 0.137 4+1 0.586 Not winning 99.27

If you play a lot of games, what is the probability to win?

Lotto - a regressive tax in Fattiges toppskatt (Norwegian text) Dagbladet, 2002-09-21, page 52

Video-lotto machines - gambling's crack-cocaine in Hard luck story The Economist, 2002-09-14, page 55

All the following documents received from
```   Date: Wed, 27 Jan 1999 16:31:15 +0100
From: "Helgesen, Rita" rita.helgesen@norsk-tipping.no
```
 LOTTO-95.DOC (MSWord Document; 24K bytes; last changed: 2006-01-13T21:47:34-CET). All lotto numbers in 1995 LOTTO-96.DOC (MSWord Document; 24K bytes; last changed: 2006-01-13T21:47:35-CET). All lotto numbers in 1996 LOTTO97.DOC (MSWord Document; 25K bytes; last changed: 2006-01-13T21:47:35-CET). All lotto numbers in 1997 LOTTO98.XLS (MSExcel Document; 31K bytes; last changed: 2006-01-13T21:47:36-CET). All lotto numbers in 1998 Lotto99.xls (MSExcel Document; 82K bytes; last changed: 2006-01-13T21:47:37-CET). Lotto numbers in 1999

Lotto belongs to the most popular gambling games in Norway. Every weekend many thousands participate in a secret hope to become this weeks Lotto millionaire. You know, Lotto millionaires are not like other millionaires.

Lotto is run by Norsk Tipping

### How to participate in the game

Go to one of the dealers, you find them at almost every corner, and fill in a lotto coupon. It consists of ten rows, each of them with numbered boxes in the range from 1 to 34. Tick of seven numbers per row, at least in two rows. Its only NOK 3.00 per row anyway. Next, give your coupon to the charming lady at the dealer's office. Now, if you really fall for the temptation to participate (probably due to the charming lady), sit down in front of your TV set at 7:25 pm next Saturday night, and tune in nrk1. Be prepared for the exciting moment when the Lotto official proudly announces that your happy moment is about to come: the lotto numbers are going to be drawn.

A big drum, containing 34 numbered balls, starts to rotate and releases randomly and one by one seven of the balls. All of it very fancy to watch. These balls are marked with the winning numbers. Next, three additional balls are drawn, containing the winning extensions. Now the exciting moment has come, compare these numbers with the numbers you have ticked off. Got all seven numbers right in one row? ...

### Statistical outcome of the game

Let's look at the probability that you get them right. But first a brief repetition of probability calculations in general.

definition:Comb(n,k) = (n!)/(k!*(n-k)!)

```If a set contains n elements, then there exist Comb(n,k)
combinations (subset) containing k elements...from the book
'Probability and Statistics for the engineering, computing and Physical
Science' by Edward R. Dougherty, 1990, page 40 ```

! mean faculty: n! = n*(n-1)*(n-2)* ...*3*2*1

ex:Comb(5,2) = (5*4*3*2*1)/(2*1*(3*2*1)) = 120/12 = 10

1. (7+0)

Your 7 numbers matches GO 7 main numbers

To answer this we must find out how many combinations you can draw 7 number out of 34

comb(34,7) = 34!/[7!*(34-7)!] = 5 379 616

You can chose to play between 5 379 616 different combinations.

Pobability(7+0) = favorable/possible = 1/5 379 616 = 0.0000001858868

You have 0.0000186 % chance of draw 7 out of the 7 main number

2. (6+1)

favorable = comb(7,6)*comb(3,1) = 7*3=21

possible = comb(34,7) = 5 379 616

Pobability(6+1) = favorable/possible = 21/5 379 616 = 0.0000039...

You have 0.00039 % chance of draw 6 out of 7 main number and 1 out of 2 extra number

3. (6+0)

favorable = comb(7,6)*comb(34-7-3,1) = 7*24 = 168

Pobability(6+0) = favorable/possible = 168/5 379 616 = 0.0000312...

You have 0.0031 % chance of draw 6 out of 7 main number.

4. (5+0) and (5+1) and (5+2)
```favorable =
comb(7,5)*[comb(34-7-3,2)*comb(3,0)+
comb(34-7-3,1)*comb(3,1)+
comb(34-7-3,0)*comb(3,2)] =
comb(7,5)*(276+72+3) =
comb(7,5)*comb(27,2) = 7371
```

Pobability(5+0,5+1,5+2) = favorable/possible = 7371/5 379 616 = 0.001370

You have 0.137 % chance of draw 5 out of 7 main number and 0,1 or 2 extra numbers

5. (4+1), (4+2) and (4+3)
```favorable =
comb(7,4)*[comb(34-7-3,2)*comb(3,1)+
comb(34-7-3,1)*comb(3,2)+
comb(34-7-3,0)*comb(3,3)] =
comb(7,4)*[828+72+1] =
comb(7,4)*901 = 31535
```

Pobability(4+1,4+2,4+3) = favorable/possible = 31535/5 379 616 = 0.005861...

You have 0.586 % chance of draw 4 out of 7 main number, and at least one extra number

NOT winning = (5379616-39096)/5379616=5340520/5379616=99.27%

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