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I am a beginner in probability and counting. I am reading an open course by MIT. While reading the introductory chapter I am stuck in one conceptual doubt, if I understand correctly an event is the subset of sample space which is the collection of all the possible outcomes now there is a very simple example of a single coin toss whose possible outcomes are:

$$\{H, T\}$$

Fine! and it makes sense now the possible events are listed as:

$$\{H, T\}, \{H\}, \{T\}, \emptyset $$

Confusion starts here, the events 2 and 3 are fine and makes sense for a fair coin i.e. the coin can be in only two states but consider the extreme case and assume that coin is still straight which may be rare but not impossible hence the event $\emptyset$ makes sense, but what about $\{H, T\}$? how can it be an event at least not in practical, am I missing something?

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    $\begingroup$ The event $\{H,T\}$ "always" occurs. Boring, but still an event. $\endgroup$ – André Nicolas Jun 5 '16 at 17:04
  • $\begingroup$ @AndréNicolas these questions seems more like facepalm but still I got my doubt cleared and so might others :) $\endgroup$ – CodeYogi Jun 5 '16 at 17:32
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    $\begingroup$ Typing \emptyset in math mode produces $\emptyset$. $\endgroup$ – N. F. Taussig Jun 5 '16 at 17:33
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    $\begingroup$ It is a reasonable question.The technical meaning of event is related to, but certainly not identical with, the everyday meaning of event. $\endgroup$ – André Nicolas Jun 5 '16 at 17:35
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    $\begingroup$ @jeremyradcliff: That would work out fine if the sample space is finite or countably infinite. However, for larger sizes, it turns out that one has to make restrictions if events are to be assigned probabilities in a nice way. $\endgroup$ – André Nicolas Jun 5 '16 at 20:53
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The event $\{H,T\}$ is the event that the coin turns up heads or tails. This event always happens and thus has probability $1$.

The empty event, $\varnothing$, should not be thought of as the event that the coin lands on its edge. It is assumed that the coin always lands heads or tails. Rather the event $\varnothing$ is the event that there is no outcome, which never happens and has probability $0$.

It might be illustrative to consider the sample space of a die roll: $\{1,2,3,4,5,6\}$. Now the event $\{2,4,6\}$ is the event that the die shows $2$, $4$, or $6$, not the event that it shows $2$, $4$, and $6$. Equivalently, $\{2,4,6\}$ is the event that the die roll is an even number, which should make sense as something we would like to consider as an event.

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Any subset of the sample space is called an event. If the sample space is finite and has $k$ elements, then there are $2^k$ different events because there are $2^k$ different subsets of a $k$-element set. In your case, $k=2$.

Perhaps this might make more sense if we look at the example of rolling a fair die. In this case, the sample space is $S=\{1,2,3,4,5,6\}$. Of course, $\{3\}$ is an event. But we can also consider the event $A$ that an odd number occurs. In this case, $A = \{1,3,5\}$, which is also a subset of $S$. Or the event that the number that shows up is a prime number. This event is the subset $\{2,3,5\}$. We can play a game and say that I win if the event "a perfect square appears" and you win otherwise. The probability of the event that a 1 or 4 occurs is $|\{1,4\}|/|S|=2/6=1/3$, so I wouldn't agree to play this game with you.

So, events are just subsets of the sample space in the mathematical sense rather than in the sense that a coin cannot come up both heads and tails (or neither heads nor tails). Consider the experiment of tossing a fair coin, with sample space $S=\{H,T\}$. This definition of the sample space automatically implies that the only possible outcomes are $H$ and $T$ - it is not possible to have both or neither occur as the outcome. Given this sample space, we can ask the question "what is the probability of the event that a head or tails appears?" This is the probability of the event $A=\{H,T\} \subseteq S$, and this probability is of course 1. If other outcomes (such as neither heads nor tails) were possible, the sample space would have to specify these other outcomes - for example, $S=\{H,T,N\}$, where $N$ denotes the possibility that the coin lands as neither heads nor tails.

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