6/49 Lotto: Probabilities, Odds, and Mathematical Expectation

Here are some hard facts about the 6/49 Lotto in terms of probabilities, odds, and mathematical expectation.

PRIZES AND PROBABILITIES

Jackpot (picking all the six winning numbers)

In choosing six out of the 49 numbers, there are C(49,6) ways. Using combination formula, this value is equal to 13,983,816. In picking all the six winning numbers, there is only one way, C(6,6). Just divide the two values obtained to determine its probability. That is, the probability of hitting the jackpot is 1/13,983,816 or 0.0000000715112. This is approximately the same as obtaining the same face 8 times in succession when tossing a fair die or getting 24 tails in succession when flipping a fair coin.
TRUE ODDS: 13,983,815 to 1

Second Prize (picking five winning numbers)

Since the five number chosen are winning numbers, the other number chosen must be from the 43 nonwinning numbers. This is equal to
C(6,5)xC(43,1) or 258. Hence, the probability of winning the second prize is 258/13,983,816 or 0.00001845. This is approximately the same as obtaining the same face 6 times in succession when tossing a fair die or getting 16 tails in succession when flipping a fair coin.
TRUE ODDS: 54,200.8 to 1

Third Prize (picking four winning numbers)

If the four numbers picked are winning numbers, the other two numbers must be from the 43 nonwinning numbers. This is equal to
C(6,4)xC(43,2) or 13, 545. The associated probability for this prize is 13,545/13,983,816 or 0.0009686, roughly the same as obtaining the same face 4 times in succession when tossing a die or 10 tails in succession when flipping a fair coin.
TRUE ODDS: 1,031.4 to 1

Fourth Prize (picking three winning numbers)

The number of ways in choosing six numbers three of which are winning numbers is equal to C(6,3)xC(43,3) or 246,820. The probability therefore of winning this prize is 246,820/13,983,816 or 0.0177, roughly the same as getting 6 tails in succession when flipping a fair coin.
TRUE ODDS: 55.7 to 1

MATHEMATICAL EXPECTATION

In computing the mathematical expectation of this game, the probability of NOT winning any of the prizes must be computed. It is obtained by subtacting the sum of the probabilities of winning at least one of the prizes from 1. Its probability is
13,723,192/13,983,816 or 0.981362455.

Aside from knowing the probability of not winning, it is likewise important to know the prizes. Assuming that the jackpot prize is Php 16,000,000, the second prize is Php 56, 000, the third prize is Php. 1,000, and the fourth prize is Php 100, then the mathematical expectation is equal to Php 16,000,000(1/13,983,816) + Php 56,000(258/13,983,816) + Php 1000(13,545/13,983,816) + Php 100(246,820/13,983,816) + 0(13,723,192/13,983,816) or Php 4.91. This means that for every 20-peso ticket, one should expect to win only Php 4.91. (From a pessimist's point of view, one should expect to lose Php 15.09)

JACKPOT PRIZE: MATHEMATICAL EXPECTATION=TICKET PRICE

The mathematical expectation computed previously assumes that the jackpot's Php 16,000,000. In lottery, however, when no one matches the winning numbers, a jackpot rolls over and is added to the next drawing. At what prize then is the mathematical expectation equal to the ticket's worth? For a ticket worth Php 20, the jackpot prize should be Php 227,001, 320. (It is assumed here that only one ticket is taken, only one wins the jackpot, and prizes are tax-free.)

FAVORABLE TIME TO BET

The favorable time to buy a ticket in 6/49 lotto, mathematically speaking, is when the jackpot prize exceeds Php 227M. This would simply mean that the mathematical expectation is higher than the ticket price.