Pure luck

Millions of us line up on Saturday nights, hoping and praying that six carefully chosen numbers from one to 42 will bring us riches galore. The truth is that no matter how much care we take in choosing those six numbers, the chance that all of them (nay, that even just one of them!) will be drawn is very, very small. First of all, I wish to make clear that this article does not intend to discourage you from betting, lest I incur the ire of the PCSO. It merely wants to inform you of the total number of ways of getting a combination of six distinct objects, in any order, from a set of 42 distinct objects. In plain English, we want to answer the question, "In how many ways can we get six numbers from 1 to 42?" More to the point, "What are the chances of my bet(s) being drawn?"

It’s not as daunting as it sounds. Think of the tambiolo (I don’t know what they call the machine that spews out the balls with numbers written on them so kindly allow me to use the well-known term, tambiolo.) Think of what happens to the 42 different balls as one by one, six of them are drawn, and not returned to the tambiolo.

Unang bola. Of course, at the very start, there are obviously 42 different choices inside the tambiolo. After one ball is drawn from/spewed out by the tambiolo, obviously, 41 balls remain.

Ikalawang bola. Now, there are 41 different choices. After another ball is drawn from/spewed out by the tambiolo, obviously, 40 balls remain.

Ikatlong bola. Now, there are 40 different choices. After another ball is drawn from/spewed out by the tambiolo, obviously, 39 balls remain.

This goes on until…

Ikaanim na bola. Now, there are 37 different choices. After another ball is drawn from/spewed out by the tambiolo, obviously, 36 balls remain.

I hope you’re still with me, because the math just starts here. The rules of counting tell us that to get the number of ways six such balls can be drawn, we must first multiply the numbers 42, 41, 40, down to 37. Thus, we get 42x41x40x39x38x37 = 3,776,965,920. Whew, almost four billion possibilities! That calls for a brief pause in the computing. (Meaning, this is not the answer yet.)

To help us catch our breath, let’s go back to the betting game. Although we may not follow any particular order in writing down our six chosen numbers, most of us encircle them (on the betting form), in ascending order (by row, left to right, top to bottom). For example, I bet on the combination 36, 37, 1, 4, 6, 5 and I encircle the numbers in the order 1, 4, 5, 6, 36, 37. Indeed, on the stub returned to me the numbers are written in that order – 1, 4, 5, 6, 36, 37. Does this alter my bet in any way? Not at all! In fact, if these six numbers do come out, maybe they won’t even come out in those two earlier arrangements. Let us say they come out in the order 4, 5, 37, 6, 1, 36. Does this mean I didn’t win? On the contrary, I still do! As long as the six numbers show up, the order in which they come out does not matter. In mathematics we call this a combination, a selection or grouping wherein order is not important.

Since we have mentioned mathematics, let us resume computation. We have just established that the arrangement 36, 37, 1, 4, 5, 6, in this discussion, is no different from 1, 4, 5, 6, 36, 37, from 4, 5, 37, 6, 1, 36, and any other arrangement of the six given numbers. However, in our initial computation (3,776,965,920, remember?) they are all accounted for. Meaning, if there are 100 possible arrangements of 1, 4, 5, 6, 36, 37 (in mathematics these are called permutations), all 100 possibilities should be counted only once; also meaning that among the 3,776,965,920, they have been double-counted 99 times! To do away with this double-count, rules of counting tell us that we should divide 3,776,965,920 by the total number of possible arrangements of the six numbers. By a procedure similar to the tambiolo draw, we get 6x5x4x3x2x1 possible ways for the six numbers to be arranged, a.k.a. six-factorial, or 720 ways. This means there were 719 double-counts! Proceeding with the required division, 3,776,965,920 / 720, we get 5,245,786. This is the answer we set out to find.

So, without further ado…

"In how many ways can we get six numbers, in any order, from 1 to 42?" In 5,245,786 ways.

"What are the chances of my bet(s) being drawn?" If you bet on one combination, one out of 5,245,786, or 0.00002 percent. If you bet on two combinations, two out of 5,245,786 or 0.00004 percent. If you bet on three combinations… If you bet on 2,622,893 combinations, 2,622,893 out of 5,245,786, or 50 percent…If you bet on all 5,245,786 possible combinations, 5,245,786 out of 5,245,786, or 100 percent!

It is inevitably true that the more you bet, the more chances you have of winning. Even if you fail, the PCSO guarantees your hard-earned money will be put to good use so more bets translate to more goodwill. (In that sense you still win.)

Some points to ponder:

Let us say you bet on all 5,245,786 possibilities, are you sure you will get back all the money you bet – more than P50 million – and the money of all the other bettors? Think again and don’t jump for joy yet. What if someone else also bet on the winning combination? Ouch!

An article on the Internet says that your chances of being struck by lighting in a given year may vary between one in 400,000 and one in 240,000, or between 0.00025 percent and 0.00042 percent. (You can boost those figures by going golfing or swimming during a thunderstorm.) That’s up to 21 times higher than your chances of winning this lottery!

Finally, no matter how small those percent figures are, no matter how close to zero they are, it is still valid to say they are not equal to zero. So can you get struck by lightning? Sure, it happens. Can you win the lottery? Sure, with pure luck.

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The author is an associate professor of Mathematics in U.P. Diliman. Her research areas include partial differential equations and operations research. A current interest is General Education Mathematics which she has been teaching for the past several semesters. E-mail her at [email protected].