Imagine you live in a world with only two types of people. Introverts and extroverts. All introverts are the same. And so are all extroverts. It's a world with only two colors, and no shades in between.

Introverts like to be alone, in our world they want to meet a friend every 14 days. Extroverts love to be around other people. In our experiment they want to hang out with a friend every 2 days.

I´s and e´s also have something in common. They have short term memory loss. Everyday at midnight their brain does a full reset. So they never remember what happened yesterday.

Now that you know the basics of our thought experiment, take a guess: Which personality type feels more alone? Introverts or extroverts?

What does this question even mean? Let's say I'm an introvert. I wake up one morning and want to meet a friend. But when I call my friends, they don’t want to hang out with me.

That day I will go to bed feeling alone. But as I said above, the next morning will be a fresh start.
 

Here's two more ways to ask the same question. Who, on average, feels alone on more days, i’s or e’s? Whose interpersonal needs are being met less, i´s or e´s?

Did you take a guess? Maybe you picked e’s 🦅. Because they need more human interaction in general. Or you picked i’s 🏴‍☠️. Because they shoot themselves in the foot by being such loners?

Let’s take a closer look at our protagonists:

Are You Going To Meet a Friend Today?

To answer this question I calculated the probability of “meeting a friend today” for each friend group mix.

What is a “friend group mix”?

• If you have one introverted and one extroverted friend, your friend group mix is 1/1, one i 🏴‍☠️ & one e 🦅.
• If you have 5 extroverted friends, your friend group mix is 0/5, zero i 🏴‍☠️ & five e 🦅.  

Each table cell (probability of meeting a friend today) is calculated like this: probability_self * sum_probability_friends.

• If an introvert has one introvert friend its: 7.14% * (1-(1-7.14%)^1) = 0.51%.
• If an introvert has two extrovert friends it’s: 7.14% * (1-(1-50%)^2) = 5.36%
• If an introvert has two extrovert and one introvert friend it’s: 7.14% * (1-((1-50%)^2 * (1-7.14%)^1)) = 5.48%

x-axis = number of extroverted friends, y-axis = number of introverted friends

x-axis = number of extroverted friends, y-axis = number of introverted friends

We can see that for extroverts the probability of “meeting a friend today” rises much faster than for introverts. Extroverts need less friends to have a high probability of “meeting a friend today”. But, if you are like me, your brain doesn't know how to work with percentages. So let's turn these % into days.

In X Days You Will Meet a Friend

Now I calculated how many days i’s and e’s have to wait until they meet a friend.

x-axis = number of extroverted friends, y-axis = number of introverted friends

x-axis = number of extroverted friends, y-axis = number of introverted friends

Think about what “kind of friends” you have. By that I mean: How many introverted and extroverted friends? Then check the charts above: How many days do you (theoretically) have to wait until you meet a friend?

Did you also get 1095 days 😅? But hold on! Where do the 1095 days for 0 friends come from?

I chose to limit the number of waiting days to a maximum of three years. Otherwise the 0/0 cell would show the number infinity for i´s and e´s. This would have messed up the charts and following calculations.

Why did I choose three years (or 1095 days)? I guess in three years everybody should be able to make and meet at least one friend.

If you haven't met a friend in three years, let's hang out quickly so my charts are correct 😁.

How Pissed Off Is Ben?

Unfortunately the previous calculations still don't tell us who feels more alone. For that we need the relative difference between what a person wants and what a person receives.

An analogy:

• Ben 👨 wants to earn 100€/h but gets only 50€/h. His salary is 50% below his expectations. 
• Lisa 👩 wants to earn 1000€/h but gets only 950€/h. Her salary is 5% below her expectations.

Who do you think is pissed off more? Ben 👨 or Lisa 👩? I think it's Ben. Because even though, just like Lisa, he earns 50€ less than what he wants.  His relative difference (50%) is much higher than Lisas relative difference (5%).

Back to our own problem: Below you can see the same “relative difference calculation” for i’s and e’s and all friend group mixes.

The “How Alone Do You Feel” Factor

x-axis = number of extroverted friends, y-axis = number of introverted friends

x-axis = number of extroverted friends, y-axis = number of introverted friends

Wow, both tables are exactly* the same. This means that introverts and extroverts are both feeling alone the same.

*The only difference is the first cell. Why is that? Earlier in the essay I told you I set a maximum waiting time of 1095 days if somebody has no friends at all 😅.

Now because this “zero friends limit” is exactly the same for i´s and e´s the relative difference for extroverts is much higher.

This means if you lock an extrovert in a single prison cell for 1095 days he will feel much more alone than an introvert. That's a no brainer. You probably already knew that.

Are You Satisfied?

I know I`m not. And since I´m the mayor of this “thought experiment world”, I'm making up one more rule: Extroverts have more friends than introverts. Which might also be true in real life?

To simplify this last chapter I calculated how alone i’s and e’s feel depending on the number of friends they have. Here’s this calculation for all possible friend group mixes:

x-axis = introvert or extrovert, y-axis = number of friends, colored cells = “feeling alone factor”

Now the numbers look different from what we saw before. Why is that? If an introvert has 1 friend, this one friend can either be an introvert or an extrovert. So the average “feeling alone factor” for one friend is: (1300% + 100%) / 2 = 700%

Introverts Feel More Alone

As you can see, the feeling alone factor for an introvert with 1 friend is higher than for an extrovert with 2 friends. We made progress. Introverts feel more alone than extroverts 🥲.  

But to be honest this feels too much like cheating. So I kept thinking: Isn't there something I might have forgotten? Yes, there is: We become friends with people who are like us. Which means extroverts have more extroverted friends & introverts have more introverted friends.

If i’s and e’s only have friends of the same kind, the new table looks like this:

x-axis = introvert or extrovert, y-axis = number of friends, colored cells = “feeling alone factor”

As you can see, introverts feel shockingly more alone 😱. If both, i’s and e’s have only one friend, introverts feel 12 times more alone. If they both have 10 friends, introverts feel over 900 times more alone.

Conclusion

• Introverts 🏴‍☠️ need some extroverts in their life to reduce feeling alone.
• Extroverts 🦅 can benefit from an introvert friend if they have social stress.

Now there's only one question left to ask: Are you an extrovert and want to hang out in 14 days?

Thanks to Maria for reading drafts of this.

Thanks @Ericf for pointing out a mistake in my calculation