# Random coincidences in life, or how it turned out that you were presented with a cake at a tractor factory

“Coincidence” is a case that seems very unlikely to us, but still happens.Have you met “coincidences” in life? In the parking lot, three red cars are standing nearby, your friend put on the exact same T-shirt for the meeting, the room with a beautiful view turned out to be the only free one, and the computer turned off the moment the guests had to open the door. We encounter situations that are themselves very unlikely. Indeed, let's see how likely my two Nissan Skylines will stop at my house? Even if there are 10,000 cars in total, and among them only two Nissan Skyline, the probability is negligible:

$$

Every time we are surprised at such “coincidences,” but is this unlikely? Let's figure it out.

First, we introduce the concept of “record an event”. This means that before the experiment (that is, before we look at the parking lot near my house) we will write on a piece of paper which particular pair of cars we want to see.

Suppose there are two parallel universes. In each of them you sit at home and are going to go to a cafe:

In the first you go and notice along the way: two identical trucks are standing at a traffic light, and two guests are celebrating their birthday at once.

Surprisingly, these coincidences are so unlikely!In the second, you are offered to first record events - try to predict "coincidences." You go to a cafe and watch the events take place.

In the first case, the probability is high, you notice all the random "coincidences".

And in the second, the probability of these events is really small: you recorded only a few “coincidences”. Most likely you will not meet any “coincidences” at all.

I explain with an example.

We will transform our universe into car parking, and the space of events into “matches” of pairs of cars, that is, into many matches of car models.

**World characteristics**

- All cars are numbered from 1 to 1,000,000.
- The number of car models: 1000, and are equally distributed.
- Parking has dimensions of 1000x1000 cars.
- A pedestrian path is laid only on one side of the parking lot.
- The distribution of cars in the parking lot is random.

Now let's walk along the track without fixing the event. What is the likelihood that we will see two identical cars standing next to each other? Pretty big.

**A little bit of math (optional)**

It's obvious that

999 will fall again - the number of movements of this pair of cars along the track

1000 - the number of models

=>

That is, despite such a large number of models, the likelihood of meeting the same adjacent cars is more than 0.6. Now, let’s fix the event:

We want to see with what probability the model K will be in the first two places.

That is, we note that it is already much smaller, or rather, it is almost impossible.

$$

- the probability that exactly model N will fall out$$

- the probability that this model999 will fall again - the number of movements of this pair of cars along the track

1000 - the number of models

=>

$$

- the probability that the first two places will be occupied by cars of the same model. However, we have 999 repetitions, that is, the probability that we still see two adjacent identical cars standing is$$

That is, despite such a large number of models, the likelihood of meeting the same adjacent cars is more than 0.6. Now, let’s fix the event:

We want to see with what probability the model K will be in the first two places.

$$

That is, we note that it is already much smaller, or rather, it is almost impossible.

What was in the spoiler: we noticed that if you walk along the track in search of the same pair of cars, we will find with a probability of more than 63%. But if you first come up with where and which cars are located, then the probability of the event will be less than one half million.

Now back to real life. Just as there were many different possible cases in the parking lot, the colors of the fur of neighboring cats, the letters of the numbers of cars at the traffic lights, and the scholarship were given at the right time can coincide in our world.

Conclusion: despite the fact that the probability of each individual “coincidence” is extremely small, the likelihood that at least some kind of “coincidence” will occur is quite high. In our parking lot, we expected at least some kind of “coincidence”, and since neither the model of the cars nor where the pair was located was important to us, the probability was high. But if you recorded an event before leaving the house, then the probability of encountering a “coincidence” is already very small. But you, of course, did not reduce the likelihood of

*this*event, you simply

*stopped taking into account the others*.

So do not be surprised when you suddenly get a cake at work. Because you would still have somehow expressed sympathy.