Can you win with 49%?

If voters are 50% likely to vote for you, you obviously have a 50% of winning the election. But what are your odds of winning if voters are only 49.9% likely to vote for you?

Let’s do the math …

The Model and Assumptions
For simplicity, assume that all voters have the same percent P of voting for you. In practice, this more resembles just the “swing” voters, and you will have different categories of voters such as your “base” that is very likely to vote for you and your opponent’s “base” that will never vote for you. But this simplification is still sufficient to illustrate the concepts.
So if P = 50%, obviously you have a 50% chance of winning the election.

But what if P drops to 49.5%?  Perhaps there’s a natural bias against you due to party, etc.  Certainly, your odds of pulling an upset and winning are still greater than 0. But it’s not still 49.5% either. So what are the odds?

Doing the math
We’ll  compute these numbers using a Monte Carlo simulation.  Source code is available at: https://github.com/MikeStall/BasicMonteCarlo
Say the district size is N. If N=5000 people, dropping P from 50% to 49.5% support means your chances of winning the election would drop from 50% to about 23%!  And when P drops to 49%, odds of victory are 7%.

Here’s a chart showing the full curve.  The horizontal axis is P (the % that an individual voter will vote for you).  The vertical axis is the % that you’ll win the overall election (assuming population size 5000.)

Picture

Note that this is not linear! Your chances of winning are not just P*N.

How does this depend on population size?
It turns out due to the Law of Large numbers, this curve gets even sharper as the population size (N) increases.  The law of large numbers means that the larger your sample size, the lower a chance of anomalies occurring. It’s easy to flip 2 heads in a row (25% odds). It’s less likely to flip 10 heads in a row. (.1% chance).   In this case, winning an election when P < 50% is “anomaly”.
Say voters are 49.9% likely to vote for you. Your odds of winning drop off rapidly as the population increases.

Picture

Summary
1. If voters are only 49.9% likely to vote for you, you still have a chance of winning the election. But it’s a steep dropoff (the blue chart).
2. The chances drop rapidly with population size (the red chart)
3. A 49% – 51% election result is actually a solid loss if the population is large.

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