Last Updated: Jan. 4, 2015

# Lottery

### On This Page

One of the most frequent questions I get is for an explanation of how to calculate the odds of the lottery. In an attempt to satisfy this quest of my viewers I shall go over the derivation of the probabilities and expected return of all the games of the Maryland lottery.

## Prerequisite Math Quiz

1. What is 4! ?

A: 4

B: 24

C: 64

D: 256

E: FOUR!!!! (said loudly)

2. You own 10 paintings but have only enough room on your walls for 4 of them. You
decide to put a different set of 4 paintings on the wall every day. How many
days can you go until running out of combinations?

A: 10

B: 24

C: 210

D: 17280

E: Paintings? I still can't tell Manet from Monet.

Click to the bottom for the answers. If you didn't get both of them right you need to go to my section on probabilities in poker and read my explanation of the factorial function and combinatorial function. This section shall assume you understand both of these. For those who got the second problem correctly the answer in this case shall be represented below by the Excel function combin(10,4).

Following are the various kinds of games offered by the Maryland lottery and their associated probabilities. The cost of a ticket for all games noted is $1 except for the lotto 6/49 with two games for $1.

## Pick 3

In Pick 3 the player chooses 2 or 3 numbers from 0 to 9. The lottery shall also
choose 3 numbers from 0 to 9 in drawings held twice a day except Sundays with only
one drawing. Each ball is chosen from a a separate machine so multiple
combinations like 3-3-3 are possible. Below are the various bets available.

**Front pair**: The player must choose only two numbers and they both must
match the first and second numbers drawn in the lottery in the correct order to win.
A winning match pays $50. The odds of getting each number correct is 1/10 so the
probability of matching 2 is (1/10)^{2}=1/100. The expected return
is $50*(1/100) = $0.50 for a house edge of 50.0%.

**Back pair**: Similar to the front pair but the player must match the
second and third numbers drawn in the lottery.

**Straight**: This is simply matching all 3 numbers in the correct order and
pays $500. The odds of getting each number correct is 1/10 so the
probability of matching 3 is (1/10)^{3}=1/1000. The expected return
is $500*(1/1000) = $0.50 for a house edge of 50.0%.

**3 way box**: The player must choose two of one number and one of another,
for example 5-5-7. If the same three numbers are drawn in the lottery, in any
order, the player wins $160. There are combin(3,2)=3 winning combinations
out of 1000 that will win (5-5-7, 5-7-5, 7-5-5). The probability
of winning is 3/1000. The expected return is $160*(3/1000) = $0.48 for a house
edge of 52%.

**6 way box**: The player must choose three different numbers,
for example 4-6-8. If the same three numbers are drawn in the lottery, in any
order, the player wins $80. There are 3!=6 winning combinations
out of 1000 that will win (4-6-8, 4-8-6, 6-4-8, 6-8-4, 8-4-6, 8-6-4). The probability
of winning is 6/1000. The expected return is $80*(6/1000) = $0.48 for a house
edge of 52%.

## Pick 4

In Pick 4 the player chooses 4 numbers from 0 to 9. The lottery shall also choose 4 numbers from 0 to 9 in drawings held twice a day except Sundays with only one drawing. Each ball is chosen from a a separate machine so multiple combinations like 3-3-3-3 are possible. Below are the various bets available.

**Straight**: This is simply matching all 4 numbers in the correct order and
pays $5000. The odds of getting each number correct is 1/10 so the
probability of matching 4 is (1/10)^{4}=1/10000. The expected return
is $5000*(1/10000) = $0.50 for a house edge of 50.0%.

**4 Way Box**: The player must choose 3 of one number and 1 of another,
for example 5-5-5-7. If the same four numbers are drawn in the lottery, in any
order, the player wins $1200. There are combin(4,3)=4 winning combinations
out of 10000 that will win. The probability
of winning is 4/10000. The expected return is $1200*(4/10000) = $0.48 for a house
edge of 52%.

**6 Way Box**: The player must choose 2 of one number and 2 of another,
for example 5-5-7-7. If the same four numbers are drawn in the lottery, in any
order, the player wins $800. There are combin(4,2)=6 winning combinations
out of 10000 that will win. The probability
of winning is 6/10000. The expected return is $800*(6/10000) = $0.48 for a house
edge of 52%.

**12 Way Box**: The player must choose 2 of one number, 1 of a second
number and 1 of a third number,
for example 5-5-7-9. If the same four numbers are drawn in the lottery, in any
order, the player wins $400. The number of winning combinations are 4!/2! =
12. This is the number of ways to order 4 numbers (4!) divided by the number of
ways to order two number (2!) since the order of the same two does not matter. The probability
of winning is 12/10000. The expected return is $400*(12/10000) = $0.48 for a house
edge of 52%.

**24 Way Box**: The player must choose 4 different numbers,
for example 2-4-6-8. If the same four numbers are drawn in the lottery, in any
order, the player wins $200. There are 4!=24 winning combinations
out of 10000 that will win. The probability
of winning is 24/10000. The expected return is $200*(24/10000) = $0.48 for a house
edge of 52%.

## Cash in Hand 7/31

The number of possible combinations the lottery will draw is combin(31,7)= 2629575.

**The general probability formula for matching x numbers out of y chosen, where there are
n to choose from is combin(y,x)*combin(n-y,y-x)/combin(n,y).** This is the number of
ways to draw x out of y correct numbers, multiplied by the number of ways to draw
y-x out of n-y incorrect numbers, divided by the number of ways to draw y balls
out of n.

The number of ways the player can draw 3 correctly is the product of the number of ways to choose 3 winners out of 7 and the number of ways to choose 4 losers out of 24. The number of ways to get 3 out of 7 matching balls is combin(7,3)=35. The number of ways to get 4 out of 24 non-matching balls is combin(24,4)=10626. So the total number of ways the player can match 3 out of the 7 numbers chosen by the lottery is 35*10626=371910. The probability of getting 3 right is thus 371910/2629575 =~ 0.1414335 .

The number of ways the player can draw 4 correctly is the product of the number of ways to choose 4 winners out of 7 and the number of ways to choose 3 losers out of 24. The number of ways to get 4 out of 7 matching balls is combin(7,4)=35. The number of ways to get 3 out of 24 non-matching balls is combin(24,3)=2024. So the total number of ways the player can match 3 out of the 7 numbers chosen by the lottery is 35*2024=70840. The probability of getting 4 right is thus 70840/2629575 =~ 0.0269397 .

The number of ways the player can draw 5 correctly is the product of the number of ways to choose 5 winners out of 7 and the number of ways to choose 2 losers out of 24. The number of ways to get 5 out of 7 matching balls is combin(7,5)=21. The number of ways to get 2 out of 24 non-matching balls is combin(24,2)=276. So the total number of ways the player can match 5 out the 7 numbers chosen by the lottery is 21*276=5796. The probability of getting 5 right is thus 5796/2629575 =~ 0.0022042 .

The number of ways the player can draw 6 correctly is the product of the number of ways to choose 6 winners out of 7 and the number of ways to choose 1 loser out of 24. The number of ways to get 6 out of 7 matching balls is combin(7,6)=7. The number of ways to get 1 out of 24 non-matching balls is combin(24,1)=24. So the total number of ways the player can match 6 out of the 7 numbers chosen by the lottery is 7*24=168. The probability of getting 6 right is thus 168/2629575 =~ 0.0000639 .

There is obviously only 1 way the player can get all 7 numbers to match. The probability of getting all 7 is 1/2629575 =~ 0.0000004 .

The following table shows each winning event, the number of combinations, the probability, the payoff, and the contribution to the expected return (the product of the probability and payoff). The player actually wins a free ticket for getting 3 right but to make things simple I shall change this to $1, the cost of a ticket.

### Cash in hand 7/31

Match | Combinations | Probability | Pays | Return |

3 | 371910 | 0.1414335 | 1 | 0.1414335 |

4 | 70840 | 0.02693971 | 4 | 0.10775886 |

5 | 5796 | 0.00220416 | 40 | 0.08816634 |

6 | 168 | 0.00006389 | 1000 | 0.06388865 |

7 | 1 | 0.00000038 | 500000 | 0.1901448 |

Total | 448715 | 0.17064164 | 0.59139215 |

From the total return in the lower right hand corner it can be seen the game pays back $0.59139215 for each $1 bet, for a house edge of 40.9%.

## Lotto 6/49

Unlike all other Maryland lottery games with lotto 6/49 the player gets two plays for $1. This table shows the prizes per game but the return is based on a per dollar bet basis. In other words the return is the product of the prize, the probability, and 2.

I will not repeat all the derivations but present the probability table below. It should be stressed that this table was for a the jackpot at a point in time and does not reflect the long term return. According to Howard Benjamin with the Maryland lottery the long term return of lotto 6/49 is about 51%.

### Lotto 6/49

Match | Combinations | Probability | Pays | Return |

3 | 246820 | 0.0176504 | 2 | 0.07060162 |

4 | 13545 | 0.00096862 | 40 | 0.07748958 |

5 | 258 | 0.00001845 | 1500 | 0.0553497 |

6 | 1 | 0.00000007 | 3650000 | 0.52203204 |

Total | 0.01863755 | 0.72547293 |

I was surprised that the return of 0.72547293 was as high as it was, about the same as keno in any casino.

## The Big Game

Below is the probability table for this game. At the time of this writing the next jackpot for March 14, 2000 was for a lump sum of 4.5 million. It should be stressed that this table was for a the jackpot at a point in time and does not reflect the long term return. According to Howard Benjamin with the Maryland lottery the long term return of lotto 6/49 is about 51%.

### The Big Game

Match | Big Money Ball | Combinations | Probability | Pays | Return |

0 | Yes | 1221759 | 0.01601774 | 1 | 0.01601774 |

1 | Yes | 744975 | 0.00976692 | 2 | 0.01953383 |

2 | Yes | 141900 | 0.00186036 | 5 | 0.00930182 |

3 | No | 346500 | 0.00454275 | 5 | 0.02271376 |

3 | Yes | 9900 | 0.00012979 | 100 | 0.01297929 |

4 | No | 7875 | 0.00010324 | 150 | 0.01548665 |

4 | Yes | 225 | 0.00000295 | 5000 | 0.01474919 |

5 | No | 35 | 0.00000046 | 150000 | 0.06882957 |

5 | Yes | 1 | 0.00000001 | 4500000 | 0.05899677 |

Total | 2473170 | 0.03242423 | 0.23860863 |

Again I was surprised by the total return, this time by how low it was at 0.23860863, for a house edge of 76.14%!

## Powerball

Please see my separate page on the Powerball Lottery.

## Annuity or Lump Sum?

As a practical example, suppose you are deciding whether to take the annuity or lump sum on a Maryland lottery jackpot. The value of the lump sum is $500,000. Maryland lottery annuities are paid over 20 years. At an interest rate of 8.8% a $1,000,000 annuity has a value of $503,752. At 9.0% the value is $497,506. Using linear interpolation we find that at an interest rate of 8.88% the value is very close to $500,000. 8.88% is much more than the interest rate of most safe investments so for this reason I would suggest opting for the annuity. The maximum tax rate of 39.6% will also apply to most of the jackpot if taken in a lump sum, as opposed to more of the payments falling in lower tax brackets if paid as an annuity.

### Lottery Annuity Values

Interest Rate | 20 Year | 25 Year | 30 Year |
---|---|---|---|

3.0% | $766190 | $717422 | $672948 |

3.2% | $753673 | $703055 | $657153 |

3.4% | $741475 | $689134 | $641930 |

3.6% | $729585 | $675641 | $627255 |

3.8% | $717996 | $662561 | $613105 |

4.0% | $706697 | $649879 | $599457 |

4.2% | $695680 | $637579 | $586290 |

4.4% | $684936 | $625650 | $573583 |

4.6% | $674458 | $614076 | $561317 |

4.8% | $664237 | $602845 | $549474 |

5.0% | $654266 | $591946 | $538036 |

5.2% | $644537 | $581365 | $526986 |

5.4% | $635043 | $571092 | $516308 |

5.6% | $625778 | $561116 | $505986 |

5.8% | $616734 | $551427 | $496008 |

6.0% | $607906 | $542014 | $486357 |

6.2% | $599286 | $532868 | $477022 |

6.4% | $590870 | $523980 | $467990 |

6.6% | $582650 | $515340 | $459248 |

6.8% | $574622 | $506941 | $450784 |

7.0% | $566780 | $498773 | $442589 |

7.2% | $559118 | $490830 | $434651 |

7.4% | $551633 | $483103 | $426961 |

7.6% | $544317 | $475586 | $419508 |

7.8% | $537168 | $468270 | $412284 |

8.0% | $530180 | $461150 | $405280 |

8.2% | $523349 | $454219 | $398488 |

8.4% | $516670 | $447471 | $391898 |

8.6% | $510139 | $440900 | $385505 |

8.8% | $503752 | $434499 | $379300 |

9.0% | $497506 | $428264 | $373276 |

9.2% | $491396 | $422189 | $367427 |

9.4% | $485418 | $416269 | $361746 |

9.6% | $479570 | $410499 | $356228 |

9.8% | $473847 | $404874 | $350865 |

10.0% | $468246 | $399390 | $345654 |

10.2% | $462764 | $394041 | $340587 |

10.4% | $457398 | $388825 | $335660 |

10.6% | $452145 | $383736 | $330868 |

10.8% | $447001 | $378771 | $326206 |

11.0% | $441965 | $373925 | $321670 |

11.2% | $437032 | $369196 | $317255 |

11.4% | $432201 | $364580 | $312958 |

11.6% | $427468 | $360073 | $308773 |

11.8% | $422832 | $355671 | $304697 |

12.0% | $418289 | $351373 | $300727 |

12.2% | $413837 | $347174 | $296858 |

12.4% | $409475 | $343071 | $293088 |

12.6% | $405199 | $339062 | $289413 |

12.8% | $401007 | $335145 | $285830 |

13.0% | $396898 | $331315 | $282336 |

13.2% | $392870 | $327572 | $278928 |

13.4% | $388919 | $323911 | $275603 |

13.6% | $385045 | $320332 | $272359 |

13.8% | $381246 | $316830 | $269192 |

14.0% | $377518 | $313405 | $266101 |

14.2% | $373862 | $310055 | $263083 |

14.4% | $370275 | $306776 | $260136 |

14.6% | $366755 | $303567 | $257257 |

14.8% | $363301 | $300426 | $254444 |

15.0% | $359912 | $297351 | $251696 |

1. B

2. C

Go back to the test.

## Internal Links

- Powerball calculator — Calculate for odds for the Powerball, or any lottery with five "white balls" and one "Power Ball."
- Mega Millions calculator — Calculate for odds for the Mega Millions, or any lottery with five "white balls" and one "Mega Ball."