# What’s Luck Got To Do With It?

What is luck? It’s a measure of how much chance you need to achieve a desired outcome, after accounting for any agency you have in affecting the outcome. If you attain the outcome you want, you realize a positive amount of luck i.e. good luck. And the more chance you needed for the outcome, the luckier you were. On the contrary, if the event does not pan out as you would have liked, you realize a negative amount of luck i.e. bad luck. And the less chance you needed for the desired outcome, the unluckier you were that it didn’t occur.

For example, imagine that the event you’re interested in is the roll of a 6-sided die. You want to roll a 4 to win a game. The probability of rolling a 4 is 1/6 and the probability of not rolling a 4 is 5/6. So the amount of chance you need is 5/6 (the probability of not rolling a 4). If you do end up rolling a 4, you rack up 5/6 or ~83% of good luck. On the other hand, if you don’t roll a 4 you suffer bad luck in the amount of -1/6.

In general, an event you want to occur has a probability ‘p’ of occurring, after fully accounting for all the controllable factors (e.g. how much effort you put into it) that affect the event. This probability may not be knowable if the event is complicated enough, like most events in life are. But the probability exists in principle. The higher the probability ‘p’, the less good luck you need and the more bad luck you suffer if the event does not happen. If the event occurs, you realize the following amount of luck: 1-p. This is the probability of the event not occurring. And if the event does not occur you realize the following amount: -p. This is the negative of the probability of the event occurring. In this measure of luck, the amount of luck in an event ranges from -1 to 1. Negative amounts correspond to bad luck and positive amounts denote good luck.

If the desired event with a probability ‘p’ of occurring is played out repeatedly, either in reality or as a thought experiment, then the average amount of luck you rack up is 0.

But hold on, you say. Most events in your life don’t play out repeatedly. What if you lock in a large amount of luck, say by winning the lottery. Then aren’t you luckier than most others? Or what if you suffered a stroke when there was a low probability of that happening. Aren’t you particularly unlucky? No. And that’s because there are a large number of events in your life and the statistics play out over all those events, leading to an average amount of luck approaching 0 for every single one of us, as the number of events approaches infinity.

To understand how the average amount of luck over a large number of events approaches 0, let’s start with a gross oversimplification: every desired event in your life has a probability ‘p’ of occurring. Then, the average luck in your life will still be 0, because the case of different events with the same probability is equivalent to the case of repeated instances of the same event as above. It doesn’t make a difference whether ‘p’ is large, which would be the case if you’re risk-averse and only wish for things that are well in your control, or ‘p’ is small as in the case of a more adventurous version of you that yearns for improbable things.

In general, there are N events in your life whose outcome you are vested in, N being large. Each event has a certain underlying probability of occurring. The average amount of luck over these N events approaches 0 as N approaches infinity. If the math of probabilities seems arcane to you, feel free to run and tweak the following simulation (in Python), and see for yourself.

import randomdef get_event_probability():

mu = 0.25

sigma = 0.1

return random.gauss(mu, sigma)num_events = 1000000

luck = 0

for i in range(num_events):

p = get_event_probability()

event_happened = random.random() < p

luck = luck + (1 - p) if event_happened else -p

print('Avg. luck: {:.2f}'.format(float(luck) / num_events))

So stop worrying about whether you’re lucky or unlucky and start focussing on more important things that you can control. Every one of us will average out to zero luck as events in our lives transpire. You can’t put your thumb on the balance scale of good and bad luck.