I suck at probability, so I thought I would ask here.

To clarify, given a population of size P, a sample size of K, and an arbitrary number of trials N, how do I compute the probability of having included each member of the population at least once in the experiment?

This problem is difficult to wrap my head around. It seems like it uses a combination of combinatorics and dependent events, which really throws me off.

Edit: This problem isn't the coupon collector's problem (please see some of my responses below). Think of the coupon collector's problem as being a special case of this problem where K = 1. My question is meant to cover an arbitrary K >= 1.

Introduction and motivation

In an effort to get more content on Tildes, I want to try and give an introduction on several 'hot topics' in semiconductor physics at a level understandable to laypeople (high school level physics background). Making physics accessible to laypeople is a much discussed topic at universities. It can be very hard to translate the professional terms into a language understandable by people outside the field. So I will take this opportunity to challenge myself to (hopefully) create an understandable introduction to interesting topics in modern physics. To this end, I will take liberties in explaining things, and not always go for full scientific accuracy, while hopefully still getting the core concepts across. If a more in-depth explanation is wanted, please ask in the comments and I will do my best to answer.

Today's topic

I will start this series with an introduction to spintronics and spin transistors.

What is spintronics?

Spintronics is named in analogy to electronics. In electronics, the flow of current (consisting of electrons) is studied. Each electron has an electric charge, and by pulling at this charge we can move electrons through wires, transistors, creating modern electronics. Spintronics also studies the flow of electrons, but it uses another property of the electrons, spin, to create new kinds of transistors.

What are transistors?

Transistors are small electronic devices that act as an on-off switch for current. We can flip this on-off switch by sending a signal to the transistor, so that the current will flow. Transistors are the basis for all computers and as such are used very widely in modern life.

What is spin?

Spin arises from quantum mechanics. However, for the purpose of explaining spin transistors we can think of an electron's spin as a bar magnet. Each electron can be thought of as a bar magnet that will align itself to a nearby magnetic field. Think of it as a compass (the electron) aligning itself to a fridge magnet when it's held near the compass.

What are spin transistors and how do they work?

Spin transistors are a type of transistor whose on-off switch is created by magnets. We take two bar magnets, whose north poles are pointed in the same way, and put them next to each other, leaving a small gap between them. This gap is filled with a material through which the electrons can move. Now we connect wires to the big bar magnets and let current (electrons!) flow through both magnets, via the gap. When the electrons go through the first magnet, their internal magnets will align themselves to the big bar magnet. However, once they are in the gap the electrons' internal magnets will start rotating and no longer point in the same direction as the big bar magnets. So that when the electrons arrive at the second magnet, they will be repelled just like when you try to push the north poles of two magnets together. This means the current will not flow, and the device is off! So, how do we get it to turn on?

By exposing the gap to an electric field, we can control the amount of rotation the electrons experience (this is called the Rashba effect). If we change the strength of this electric field so that the electrons will make exactly one full rotation while crossing the gap, then by the time they reach the second big bar magnet they will once again be pointing in the right direction. Now the electrons are able to move through the second big bar magnet, and out its other end. So by turning this electric field on, the spin transistor will let current flow, and if we turn the electric field off, no current will flow. We have created an on-off switch using magnets and spin!

That's cool, but why go through the effort of doing this when we have perfectly fine electronics already?

The process of switching between the on and off states of these spin transistors is a lot more energy efficient than with regular transistors. These types of transistors leak a lot less too. Normal transistors will leak, meaning that a small amount of current will go through even when the transistor is off. With spin transistors, this leak is a lot smaller. This once again improves the energy efficiency of these devices. So in short, spin transistors will make your computer more energy efficient. This type of transistor can also be made smaller than normal transistors, which leads to more powerful computers.

Feedback and interest

As I mentioned, I wrote this post as a challenge to myself to explain modern physics to laypeople. Please let me know where I succeeded and where I failed. Also let me know if you like this type of content and if I should continue posting other similar topics in the same format.

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