ON Monday, March 29, 2021, it was reported that a lone bettor from Pagadian City, Zamboanga del Sur won the jackpot in the 6/55 Grand Lotto draw held on Saturday, March 27, 2021. The player picked the winning six-number combination of 38-35-11-22-39-47, which had a total jackpot prize of P298.77 million.
Indeed, this news item can serve as a good review of basic probability concepts. There are some interesting questions that people can try to answer. One, does the jackpot worth P298.77 million make the game fair? Two, if it is not a fair game, then what amount will make it fair? Three, where else should people consider putting their money?
The game has the numbers 1 to 55, and it requires a person to randomly pick six numbers out of these. Selection is done without replacement. This means that once a number has been picked, it can no longer be returned to the pool.
Also, in this game, the arrangement of the numbers does not matter. This means that people are dealing with combinations rather than permutations. So, if six numbers are to be picked randomly from 55, then there are 28,989,675 possible combinations, and only one can win. Each lotto ticket costs P20 and represents one bet. Now, if the game were to look for a winning permutation instead of combination, then 20,872,566,000 possible permutations would have to be considered, and the probability of winning would drop significantly.
By definition, expected value is a weighted average where the possible values of a random variable are weighted by their corresponding probabilities of occurrence. A fair game is one whose expected value is zero. If the 6/55 Grand Lotto were a fair game, then neither the bettor nor the lotto operator would be favored.
So, here are the answers to the questions posed earlier. Regarding the expected value of the 6/55 Grand Lotto, it is computed as follows: (P298,770,000.00)(1/28,989,675) + (-P20)(28,989,674/28,989,675) ≈ -P9.69. P298,770,000 is the jackpot prize, and it is weighted by its corresponding probability of occurrence, which is 1/28,989,675. The -P20 represents the amount of money that a person pays the lotto operator in every transaction, and it is weighted by its corresponding probability of occurrence, which is 28,989,674/28,989,675. Over a very large number of bets, the bettor, therefore, stands to lose about P9.69 on average to the lotto operator.
Clearly, this is not a fair game because the lotto operator is being favored. This should not come as a surprise, though, since lotto outlets are meant to raise funds for charity. Also, to many people, P298.77 million might already look obscenely huge; nevertheless, in reality, this jackpot amount is not yet huge enough to make the game fair.
So, to answer the second question regarding the jackpot amount that will make the game fair, people can set up an algebraic equation and then solve for the unknown value. (X)(1/28,989,675) + (-P20)(28,989,674/28,989,675) = 0. Isolating X on one side of the equation yields thus: X = ((-P20)(-28,989,674/28,989,675))/(1/28,989,675) ≈ P579,793,480.00.
If one wishes to surely win, then he should have pockets deep enough to churn out this amount of capital: (P20 per bet)(28,989,675 possible bets) = P579,793,500. Again, if P298,770,000 is the jackpot amount, then it will be a losing venture. The jackpot amount should exceed the required capital to make it worthwhile for the bettor.
To answer the last question regarding where else to put one’s money, one should be aware of some practical considerations. Of course, it is unlikely that the average Pinoy can make 28,989,675 bets in one day. It is also unlikely that the average Pinoy has close to P580 million in capital. It is also unlikely that the average Pinoy can win the 6/55 Grand Lotto.
What is certain, though, is that this world is finite. On a microeconomic level, the likelihood of getting sick and dying far outweighs the likelihood of winning the 6/55 Grand Lotto. On a macroeconomic level, experiencing financial distress is made highly probable by a large informal economy, a low propensity to save and invest, and widespread financial exclusion. With people losing loved ones and the entire economy suffering, this pandemic should make each person realize his finitude all the more.
Those fortunate enough to have buffer resources during this pandemic can reflect on two options. One option is to become someone who generously spends P20 per lotto ticket per day for an outcome that is unlikely to happen, although the money supposedly goes to charity. Another option is to become someone who saves money, pays for insurance, and makes wise investments, in anticipation of life events that are far more likely, or even certain, to happen.
Where should charity begin?
Dr. Ser Percival K. Peña-Reyes teaches economics at the Ateneo de Manila University.