Hot Lotto

Revised 2/6/08

In the multi-state Hot Lotto Lottery five balls are chosen at random without replacement from a container containing 39 balls, numbered 1 to 39, and then a sixth ball, the hot ball, is chosen from a second container containing 19 balls, numbered 1 to 19. To play, a person chooses five numbers and a hot ball number. Payoffs are made depending on how many numbers match the player's choices and whether or not the hot ball number is matched, according to the following schedule for a $1 bet, and have the indicated probabilities of occurring: Hot Lotto Odds

Matches - - -Payoff - - - - - - - Probability - - - - Approximately

5W 1R - - Jackpot - - - - - - - 1 / 10,939,383 - - - 1 / 10,939,383

5W 0R - - $10,000 - - - - - - 18 / 10,939,383 - - - 1 / 607,744

4W 1R - - - $500 - - - - - - -170 / 10,939,383 - - - 1 / 64,349

4W 0R - - - - $50 - - - - - 3,060 / 10,939,383 - - - 1 / 3,575

3W 1R - - - - $50 - - - - - 5,610 / 10,939,383 - - - 1 / 1,950

3W 0R - - - - - $4 - - - 100,980 / 10,939,383 - - - 1 / 108

2W 1R - - - - - $4 - - - - 59,840 / 10,939,383 - - - 1 / 183

1W 1R - - - - - $3 - - - 231,880 / 10,939,383 - - - 1 / 47.2

0W 1R - - - - - $2 - - - 278,256 / 10,939,383 - - - 1 / 39.3

Total chance of winning = 679,815 / 10,939,383, or about 1 / 16.1

2W 0R - - - - - $0 - - 1,077,120 / 10,939,383 - - - 1 / 10.2

1W 0R - - - - - $0 - - 4,173,840 / 10,939,383 - - - 1 / 2.62

0W 0R - - - - - $0 - - 5,008,608 / 10,939,383 - - - 1 / 2.18

For example, the probability of getting 4W 0R is computed as follows: From combinatorial theory, there are 39!/5!34! = 575,757 ways to choose 5 numbers from a group of 39. There are 19 ways to choose 1 number from a group of 19. Therefore there are 19 × 575,757 = 10,939,383 ways of making a hot ball selection. (It is reasonable to assume that when the drawing is held each of these 10,939,383 ways is equally likely to occur.) The number of favorable ways of getting exactly 4W out of one's 5 choices is (5!/4!1!) × (34!/1!33!) = 5 × 34 = 170; the number of favorable ways of getting exactly 0R out of one's 1 choice is 18. Therefore the total number of ways of getting exactly 4W 0R is 170 × 18 = 3,060, and thus the probability of getting 4W 0R is 3,060/10,939,383.

The minimum Jackpot is $1,000,000. However, this amount could be paid in 25 yearly payments and so would not have a 'present value' nearly that large ($500,000 would be a reasonable estimate currently, and this amount could be taken out in a lump sum). Also, one must pay taxes on one's winnings, which would further decrease the amount one would actually win. There is also the possibility that more than one person would win, in which case the Jackpot would be split evenly among all winners* (this does not apply to any of the other prizes).

*See the movie, Bruce Almighty.

In general, one is less interested in the probability of winning than in how much one on average could 'expect' to win when one gambles. To compute the expected payoff on a bet one multiplies the amount of each payoff (prize) times the probability (odds) of getting that payoff, and adds these together. Note that if one ignores the Jackpot (See Reasonable Expectation), which one has a chance of less than one in 10 million of winning (one's chances of being hit by lightning are much greater), one's expected payoff on a $1 bet is only 23.7 cents!:

(10,000 × 18 + 500 × 170 + 50 × 3,060 + 50 × 5,610 + 4 × 100,980 + 4 × 59,840 + 3 × 231,880 + 2 × 278,256 + 0 × 1,077,120 + 0 × 4,173,840 + 0 × 5,008,608) / 10,939,383 = 2,593,932/10,939,383 = .237

One could increase one's expected payoff by using the Sizzler option. For an additional $1 the expected payoff would be:

(30,000 × 18 + 1500 × 170 + 150 × 3,060 + 150 × 5,610 + 12 × 100,980 + 12 × 59,840 + 9 × 231,880 + 6 × 278,256)/10,939,383 = .711 (which is 3 × .237)

Since this costs $2, the expected return is 35.6 cents per dollar, still a very poor payoff.

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