TITLE: what is difference between probability and probability space?
QUESTION [6 upvotes]: I am a beginner in probability and started reading the relative material. I encountered the exercise question "find the probability space for tossing a fair coin till the first head is observed". so, I would like to ask you, are probability and probability space different? If yes, please provide the example with the above as the question.

REPLY [3 votes]: I will try to explain it in very simple words:
Finding the probability space means finding each possible event and its probability.
Let's do this with regards to the specific question at hand:

Event #$1$:

You toss the coin once and it lands on head
The probability for this event is $\frac{1}{2}$

Event #$2$:

You toss the coin once and it lands on tail
You toss the coin again and it lands on head
The probability for this event is $\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}$

$\dots$
Event #$k$:

You toss the coin $k-1$ times and it lands on tail each time
You toss the coin again and it lands on head
The probability for this event is $\frac{1}{2^{k-1}}\cdot\frac{1}{2}=\frac{1}{2^{k}}$

$\dots$

Of course, you have an infinite number of events in this probability space.
You may also assert that the sum of the probabilities of these events is equal to $1$:
$$\sum\limits_{n=1}^{\infty}P(\text{number of tosses}=n)= \sum\limits_{n=1}^{\infty}\frac{1}{2^{n}}=1$$