9.9% -- Chance of 84 Successive No-Winner Grand Lotto Draws

According to the Philippine Charity Sweepstakes Office the Jackpot Prize that someone could've won during the 84th Grand Lotto Draw the other day was a whopping 693 million pesos. During the upcoming Draw tonight, the Jackpot is likely to exceed 700 million pesos.  But how likely was this RUN of 84 successive No Winner Draws of the Grand Lotto?  What was the mathematical probability of 84 No Winner Grand Lotto Draws?  I think I can answer this question with a good estimate based on some numbers mentioned by Margie Juico of the PCSO on TV this morning, specifically that there have been 84 Draws since this run began.

First of all, there are 28, 989, 675 possible six number combinations in Grand Lotto 6/55.  

C(6,55)=55!/(6!(49!))=55*54*53*52*51*50*(49!)/(6*5*4*3*2*1)*(49!)=28,989,675

The chance that on any given Draw there will be NO WINNER--depends on what percentage of these nearly 29 million possible combinations are actually placed as 20 peso bets by the Public.  This percentage varies from Draw to Draw. For example when it started as a P30 Million jackpot there were probably much fewer bettors than tonight's Monday Draw which could award a jackpot exceeding P700 Million. 

But what percentage of the 29 million possible 6 number combinations actually ARE placed as bets?

We can estimate the AVERAGE  number of bets placed during each of the last 84 draws from some things that PCSO's chair MARGE JUICO  told ABSCBN's Karen Davila on Headstart this morning.  She said that for each peso bet, 15% goes to PCSO, 30% to Charity and 55% to Jackpots (less 2% to the Operators!). Since the Jackpot is now about P700M after 84 successive no winner Draws (3 times a week since last May), this implies a total of (700million/53%) or 1.32 Billion pesos has been bet since May. Assuming there is negligible duplication of 20 peso bets this means that an AVERAGE of about 780,000 unique bets have been placed during each of the last 84 GrandLotto Draws.  

1.32 billion/(20*84) = 786,163 bets on an "average Draw"

Even though there may now be millions of bets being placed, there were surely far less during the early draws when the Jackpot was much smaller.  

If only 786,163 unique bets out of 28,989,765 are placed on a given Draw, the probability that there will be a winner is

786,16/28,989,765 = 2.7% is the chance of a winner being drawn or 97.3% is the chance of no winner during each draw.

So what are the chances that 84 times in a row, there would be NO WINNER of the Grand Lotto Jackpot? The answer of course is 97.3% raised to the 84th power or 

.973 multiplied by itself 84 times or a slighty improbable 9.9%