Foundations — Probability basics — sample space, events, P(E) = favourable - total

This page assumes you know nothing about the notation. We build every symbol from a picture, one at a time, in the order the topic needs them.

1. The experiment and its "outcomes"

Before any symbol, there is a thing you do that has an uncertain result: flip a coin, roll a die, pick a marble. We call this an experiment.

Picture a coin lying on a table after the flip. It is either Heads-up or Tails-up. Each of those two final pictures is one outcome.

Why the topic needs it: everything else — sample space, events, probability — is built by collecting outcomes into groups. If we cannot name the individual results, we cannot count them, and probability is counting.

2. Sets and the curly braces $\{\}$

The parent note writes $S=\{\text{H},\text{T}\}$. Those curly braces are the first real notation.

Picture a basket. Whatever you drop in is "in the set". Two rules the picture makes obvious:

• A thing is either in the basket or not — no half-membership.
• Order and repeats don't matter: $\{\text{H},\text{T}\}$ is the same basket as $\{\text{T},\text{H}\}$.

Why the topic needs it: the sample space and every event are sets. Braces are the container the topic never stops using.

3. The sample space $S$ (and its twin $\Omega$)

Picture a board with every final photo of the experiment pinned to it. For one die that is six photos: the faces $1,2,3,4,5,6$.

Why the topic needs it: $S$ is the denominator of the whole subject. See Sample Space for how to build big ones systematically.

4. The counting symbol $n()$

The parent writes $n(S)=6$ and $n(E)=2$.

Picture yourself pointing at each photo and saying a number — the last number you say is $n$.

Why the topic needs it: probability is a fraction of counts. $n()$ is the tool that turns a basket of pictures into a number you can divide.

5. Events $E$ — circling part of the board

Picture drawing a loop around some of the pinned photos. Everything inside the loop is "the event happens"; everything outside is "it doesn't".

Two extreme loops the topic names specially:

• Loop around exactly one photo → simple event, e.g. $\{4\}$.
• Loop around several photos → compound event, e.g. $\{2,4,6\}$.
• Loop around the whole board → the sure event, which is just $S$ itself.
• Loop around nothing → the impossible event, written $\varnothing$ (next section).

Why the topic needs it: the question "what's the chance of an even roll?" is the question "how big is the loop $\{2,4,6\}$ compared to the whole board?". Events turn a word-question into a countable set. See Events and Set Theory.

6. The empty set $\varnothing$

Picture an empty loop on the board. "Roll a 7 on a normal die" describes it: no photo qualifies, so the loop catches nothing.

Why the topic needs it: it is the outcome-set of anything impossible, and impossible must get probability $0$. Without a symbol for "nothing", the formula would have no way to say "never".

7. Complement — everything NOT in the loop, symbol $E^{\prime }$

Picture the same board and loop: $E$ is inside the loop, $E^{\prime }$ is the entire shaded region outside it. Together they tile the whole board with no gap and no overlap.

Why the topic needs it: counting "everything except" is often far easier than counting the thing itself, which is exactly why the parent's Complement Rule $P(E^{\prime })=1-P(E)$ exists. See Complement of an Event.

8. Union $\cup$, intersection $\cap$, and "no overlap"

The parent's Mistake 2 lives entirely in these two symbols.

Picture two overlapping loops. The union is the whole shaded blob (both loops merged). The intersection is the little almond where they cross.

Why the topic needs it: "or" questions are unions; the double-counting mistake happens because $\cup$ merges overlaps automatically but naive addition does not. See Mutually Exclusive Events.

9. The probability symbol $P(E)$ and the ratio

Now every ingredient exists, so the headline formula finally makes sense symbol-by-symbol.

Reading the extremes:

• Empty loop: $P(\varnothing )=\frac{0}{n(S)}=0$ — impossible.
• Whole board: $P(S)=\frac{n(S)}{n(S)}=1$ — certain.
• Any loop in between: $0\le P(E)\le 1$, because a part can't be smaller than nothing or bigger than the whole.

How the foundations feed the topic

Equipment checklist

Cover the right side and answer each before moving on.