Sanderson M. Smith

A STATISTICAL ANALYSIS OF CALIFORNIA'S
SuperLOTTO Plus

SuperLOTTO Plus was introduced by the California Lottery on June 4, 2000. There are nine ways to win, and the jackpot starts at seven million dollars.

The rules for SuperLOTTO Plus are relatively simple. There are two number sets:

Set A: The integers from 1 to 47 inclusive.

Set B: The integers from 1 to 27 inclusive.

A player spends \$1 for the privilege of selecting five numbers from Set A, and one number, called the MEGA number, from Set B. All prize payout amounts are pari-mutual. That is, the amount of money varies from week to week, depending upon the sales levels and the number of winners. The jackpot prize, obtained when one matches all five numbers and the MEGA number, has a stated minimum value of seven million dollars.

There are nine ways to win in SuperLOTTO Plus. Probabilities and odds associated with the game are displayed in the table below. Here are some notes and observations relating to the table:

• When playing, two events are involved. One involves choosing five numbers from Set A, and the second consists of picking one number from Set B. The two events are statistically independent, so related probabilities can be multiplied, as is done in the table.
• The probability that a player matches the MEGA number is 1/27, and the probability that the MEGA number is not matched is 26/27.
• The symbolism nCr represents the number of different sets of r objects that can be selected from a set of n objects. For instance, consider the set {red,white,blue,green}. If I want to choose two colors from this set, there are 4C2 = 6 ways this can be done. The six possible sets are {red,white}, {red,blue}, {red,green}, {white,blue}, {white,green}, and {blue,green}. Modern calculators (the TI-83, for instance), have menus that include nCr.
• A number such as 2.41451E-08 is 2.41451/108, which is 0.0000000241451. (This number does appear in the table.)

It is often helpful to represent very small probabilities in the form 1/x using the algebraic identity

x = 1/(1/x), if x is not zero.

For instance, the probability of hitting the jackpot is 0.0000000241451 (see table). Using the algebraic identity, this can be written as

0.0000000241451 = 1/(1/0.0000000241451) = 1/41,416,353.

In other words, the odds of winning the jackpot are 1 in 41,416,353.

Here is a statistical analysis of SuperLOTTO Plus:

 EVENT SYMBOLIC PROBABILITY NUMERICAL PROBABILITY ODDS: 1 in -- Match FIVE and MEGA (5C5)(42C0)(1/27)/47C5 2.41451E-08 41,416,353 Match FIVE and NO MEGA (5C5)(42C0)(26/27)/47C5 6.27771E-07 1,592,937 Match FOUR and MEGA (5C4)(42C1)(1/27)/47C5 5.07046E-06 197,221 Match FOUR and NO MEGA (5C4)(42C1)(26/27)/47C5 0.000131832 7,585 Match THREE and MEGA (5C3)(42C2)(1/27)/47C5 0.000207889 4,810 Match THREE and NO MEGA (5C3)(42C2)(26/27)/47C5 0.005405111 185 Match TWO and MEGA (5C2)(42C3)(1/27)/47C5 0.002771852 361 Match ONE and MEGA (5C1)(42C4)(1/27)/47C5 0.013512778 74 Match NONE and MEGA (5C0)(42C5)(1/27)/47C5 0.020539423 49 TOTAL PROBABILITY OF WINNING SOMETHING = 0.0425746088 0.042574608 = 1/(1/0.042574608) = 1/23.4881786. Overall odds of winning are 1 in 23

As the table indicates, the probability of hitting the jackpot is quite small. To use a time perspective, suppose you continually "invested" \$1 per second in SuperLOTTO Plus. Mathematically speaking, you would have to play continuously for over 15 months to expect to hit the jackpot once. And, you could expect to invest over 41 million dollars to get a return that could be as small as 7 million dollars.

But then, while they may be few and far between, there are those who spend \$1 and end up as big winners in games like SuperLOTTO Plus.

"Oh, many a shaft at random sent

Finds mark the archer little meant."

-Sir Walter Scott: The Lord of the Isles