# Lucky for Life: Odds and How to Play

TR Smith is a product designer and former teacher who uses math in her work every day.

Lucky for Life originated as a Connecticut-only lottery in 2009, and in subsequent years became a multi-state lottery among six states in New England. In 2016 the game will expand to include at least 21 states spread across the US.

What sets Lucky for Life apart from other lotteries is that it offers two cash-for-life prizes, one grand prize and one second prize. If all your numbers match the results of the drawing, you win the grand prize of \$1000 a day (\$7000 a week) for the rest of your life. If you make the next-best partial match, you win the second prize of \$25,000 a year for the rest of your life, with a guaranteed minimum of 20 annual payments.

The matrix of numbers has changed over the years. When Lucky for Life was first created, players chose 4 white balls and 1 green ball, hence the original name Lucky4Life. In 2012, players chose 5 balls white from 1 to 40 and a lucky green ball from 1 to 21. Then in 2013 the game was changed to 5 white balls drawn from between 1 and 43 and a lucky ball drawn from between 1 and 43. Since 2015, players select 5 white balls from 1 to 48 and a lucky ball from 1 to 18.

## How to Play

Each Lucky for Life lotto slip costs \$2 per play. A play consists of choosing 5 distinct numbers from a field of 1 to 48, and then a separate "lucky number" from 1 to 18. This lucky number may be equal to one of the 5 previous numbers. The grand prize is awarded to the ticket holder who has a perfect match of 5/5 regular numbers plus the lucky number. Lower prizes are awarded for partial matches, just as in other lotteries.

## Probability of Winning Lucky for Life

The probability of matching 5/5 regular numbers and the lucky number is

1/[(48 C 5)*18]
= 1 / 30821472
= 0.000000032445
= 0.0000032445%
= odds of 1 in 30,821,472

In this calculation, (48 C 5) is the combination function "48 choose 5," aka the number of ways to choose 5 distinct objects from a set of 48 without replacement. In the denominator there is a factor of 18 since there are also 18 ways to choose the lucky number. With odds of about 1 in 30.8 million, the Lucky for Life lottery offers a better chance of winning than other multi-state lotteries such as Powerball or Mega Millions.

The probability of your ticket matching 5/5 numbers without the lucky number is

17 / 30821472
= 0.00000055156
= 0.000055156%
= odds of about 1 in 1,813,027.76

The probability of matching 4/5 numbers plus the lucky number is

5*43 / 30821472
= 0.00000697565
= 0.000697565%
= odds of 1 in 143,355.68

Here are the odds of all the various matches that are awarded prizes:

Match
Odds
Prize
5/5 + lucky
1 in 30,821,472
\$1,000 a day (\$7000 a week) for life
5/5
1 in 1,813,028
\$25,000 a year for life
4/5 + lucky
1 in 143,356
\$5000
4/5
1 in 8,433
\$200
3/5 + lucky
1 in 3,413
\$150
3/5
1 in 201
\$20
2/5 + lucky
1 in 250
\$25
2/5
1 in 15
\$3
1/5 + lucky
1 in 50
\$6
lucky number only
1 in 32
\$4
Overall odds of winning any prize
1 in 7.8

The smallest prize, \$3, is awarded for a 2/5 match without the lucky number, since that partial match has the highest probability of occurring. The \$5,000 prize for matching 4/5 plus the lucky number may be awarded on a pari-mutuel basis, meaning it could vary according to the prize funds available and number of winners. According to current lottery rules, winners of the third prize level are guaranteed \$5,000 when there are up to 1,000 winners. If there happen to be more than 1000 winners, they will all split \$5,000,000 equally.

The two top prizes will also be split among more than one winner if there happen to be several winning tickets sold.

Why is the probability for matching only the lucky number 1 out of 32, rather than 1 out of 18? The answer is because you are calculating the probability of matching only the lucky number and 0 out of 5 regular numbers. Because this is a more specific event, the probability is smaller.

## Taking the Cash Option

Winners who don't want to receive the weekly or annual payments can opt for a lump sum prize instead. The amount of the lump sum is less than the sum of all the future payments because the future payments include interest earned from the lottery's investments. Currently he lump sum option for the grand prize is \$5,750,000, while the lump sum option for the second prize is \$390,000. These amounts will be divided by the number of winners if there are several.

Should you take the cash option if you win the lottery? The answer depends on whether you can invest the money at a rate of return that exceeds the lottery's investments. In the US, lotteries guarantee their annuity payments by investing in government-backed securities. Which ever option you decide, don't forget to factor in your state and federal tax burdens.

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