Does My Vote Count? Yes It Does!

I think no one was shocked by BN's victory in Hulu Selangor yesterday - or at least that's what I believe. I mean, look at P. Kamalanathan's electric smile, and add that to Najib's charms and promises - the 1725 majority won by the ruling coalition in an opposition state came as no surprise for me.

Let's talk about election in general. Does democracy provide each voter the chance to decide the fate of it's country? Or is this one-man-one-vote system inherently fails to fulfil its promise to give equal opportunity for each and every individual citizens? Skeptics would tell us not to vote because, so they believe, in reality no single voter has the power to elect a government. In a way, the skeptics are right: as individuals - each of us is just one meaningless voter - drowned by the overwhelming thousands of others in our constituencies. (In the context of Hulu Selangor by-election, how futile one-vote seemed when it's matched to 48999 others)

But hold on a moment. Last night as I pondered upon these questions, I thought of using measures of random probability to try to resolve this great debate on the relevance of our democratic system. And I've done it! Sadly, the skeptics are right!

From my calculation of the probability (or chance) that our single vote can have an impact to an election, I've found out that it actually matters really less.  In fact, there is very little chance that my single vote could determine the winning candidate in an election. In other words, my vote is unlikely to affect the outcome. (Obviously, this calculation must take into account many assumptions. The details of this calculation is included later in this post. I really recommend you to see it - I was astonished myself, because earlier I got to a wrong conclusion in an opposite direction). Maybe it sounds obvious to you, but do allow me to explain it further.

Yes, if we don't vote, we practically have no influence on the outcome - maybe because we just don't care. But if we do care and we go out to vote, what difference does it make? In a two party election - even without our vote - it is very likely that any one of the party wins the election anyway. So is it worthwile to go out and vote?

And here come the calculations, or a thought experiment:

But first let's discuss several assumptions about this calculation: (1) There are only two contesting parties (to make calculations easier). Maybe later I'll try to calculate how much an extra candidate changes these figures. (2) For each individual voter, there is 1/2 chance that he/she votes for any one party. (3) The votes are random, and the candidates are equally matched (not like Barack Obama vs Mat Selamat 'the terrorist'). (4) Each voter does not affect the vote of another. We can never be sure of which party any single person would vote as there are too many external and internal factors at play when someone decides whom to vote for.

Let's start with an election with only 2 voters, and  2 candidates: 'A' and 'B'.

The probability that both voters vote for 'A':  0.5 x 0.5 = 0.25

The probability that both voters vote for different candidates: 0.25+0.25 = 0.5

The probability that both voters vote for 'B':  0.5 x 0.5 = 0.25

Now let's assume the election has 3 voters and you are the third voter. What difference does your vote make?

The probability that the other two voted for 'A': 0.5 x 0.5 = 0.25

The probability that the other two voted  for different candidates: 0.25+0.25 = 0.5

The probability that the other two voted for 'B': 0.5 x 0.5 = 0.25

This means that there is 1/2 chance that there is a tie without your vote, and so your vote would eventually determine who's the winner. Also, do notice that even without your vote, each candidate has a 1/4 chance of winning.

Let's move on to a larger population of voters: 4 voters.

The probability that all 4 voted for 'A': 0.5 x 0.5 x 0.5 x 0.5 = 0.0625

The probability that only 3 voted for 'A':  0.25

The probability that both candidates has 2 votes each (a tie): 0.375

What if you are the fifth voter. Again, using the calculations above, the chance of your vote making a change in the election result is 0.375 (when the other 4 voters voted to a tie). Notice that the probability is getting smaller: from 0.5 (for 3 voters) to 0.375 (for 5 voters). What happens eventually is when the number of voters rise, the probability figure gets smaller - but by how much? Does it eventually come down to zero? Almost.

The probability that your vote is the winning vote, if you are:

the 3rd voter: 0.5 (1/2)

the 5th voter: 0.375 (~1/3)

the 11th voter: ~ 0.2461 (1/4)

the 21st voter: ~ 0.1762 (1/6)

the 101th voter: ~0.0796 (1/13)

The formula is as follows:

provided there are 2 candidates, for (v+1) number of voters, and you are the (v+1)th voter, the probability of an equal vote for the two parties, excluding your vote = [vC(v/2)] x [0.5 ^(v)]

As you can see, the reduction is consistent as the number of voters increases and soon the chances become more insignificant - the probability would drop even further when there are thousands of voters. Consider if you were the 101th voter, according to my formula, there is only about 1/13 chance that your vote will decide the winner. Simply put, in 12 out of 13 times you vote (in an electorate with 101 voters), either one of the contesting parties would win anyway - regardless of your vote.

This is quite insightful, I guess. Is it any wonder, then, that there are those who contend that their vote doesn't count. They'd rather watch Sunday TV shows or go out for a movie, than queuing up at an electoral booth. It seems to them that whether or not they came out to vote, the majority of other voters would do so, hence their vote couldn't have much of an impact to the outcome. In a way, they are right: the majority of others would usually show up to vote (76% in Hulu Selangor).

So are the candidates just wasting huge amounts of resources on grandeur election campaigns? And are the voters better off doing other things on election day?

Considering what is at stake here - electing our national governing body - even a 1/1000 chance of making a difference should make each try a worth attempt. So don't forget to go out to vote next time. It may be the 1/1000 lucky strike that your vote could actually make a difference!