Solution. There are two outcomes when a coin is tossed: heads ($H$) or tails ($T$). Therefore, the sample space for tossing a coin once is $\Omega=\{H,T\}$. The sample space for tossing a coin twice is $\Omega=\{HH, HT, TH, HH\}$.

Solution. The sample space of rolling a dice is:

The number of sample points is $6$, that is, there are $6$ possible outcomes in total.

Solution. Here is a list of some events that may be of interest:

• event $A$ - observe an odd number, that is, $\{1,3,5\}$.

• event $B$ - observe a number greater than $3$, that is, $\{4,5,6\}$.

• event $C$ - observe a $1$, that is, $\{1\}$.

• event $D$ - observe a $6$, that is, $\{6\}$.

Here, event $A$ occurs when we observe a $1$, $3$ or $5$. Since multiple outcomes satisfy event $A$, it is considered to be a compound event. The same can be said for event $B$.

On the other hand, events $C$ and $D$ are called simple events because they cannot be decomposed, that is, there is only one way in which these events happen. We sometimes denote simple events using the letter $E$ - for instance:

• simple event $E_1$ - observe a $1$.

• simple event $E_2$ - observe a $2$.

• and so on.

Notice that any event, whether it be compound or simple, is a subset of the sample space:

Just like for sets, we can visualize events using a Venn diagram:

Recall that the compound event $B$ occurs when we observe a number greater than $3$, that is:

We can visualize event $B$ with a Venn diagram:

Solution. Some examples of events we might be interested in are:

• event $A$ - getting all heads.

• event $B$ - getting exactly one tails.

• event $C$ - getting more than one tails.

• event $D$ - getting at least two heads.

Note the following:

• event $A$ is a simple event because there is only one way in which this event occurs $(\mathrm{HHH})$.

• event $B$ is a compound event because this outcome can occur in multiple ways $\mathrm{TTH}$, $\mathrm{HTH}$ or $\mathrm{HHT}$.

• events $C$ and $D$ are also compound events. For instance, event $C$ occurs when we either get two or three tails.

Intuition. The first two axioms of probability are straight-forward. Axiom 1 states that the probability of an event cannot be negative. If $\mathbb{P}(A)=0$, then the event cannot occur.

Axiom 2 states that the probability of the sample space $\Omega$ is equal to one, that is, we must observe an outcome contained in the sample space. This should make sense because the sample space by definition must contain all possible outcomes, and so one of the outcomes in the sample space must occur.

Axiom 3 is much less intuitive. This axiom states that if events are disjoint or mutually exclusive, then the probability of their union must be equal to the sum of the probabilities of each event. For example, suppose we roll a fair dice once. The sample space is $\Omega=\{1,2,3,4,5,6\}$. Let's define events $E_1$ and $E_2$ as follows:

• event $E_1$ - observe a $1$.

• event $E_2$ - observe a $2$.

These events are mutually exclusive, that is, they cannot happen at the same time. Axiom 3 states that the probability that either events $E_1$ or $E_2$ happens is the sum of the probabilities of each event:

Let's understand what this intuitively means. For a fair dice, events $E_1$ and $E_2$ each occur with a probability of $1/6$. The probability that either $E_1$ or $E_2$ occurs should be the sum of their probabilities:

What we have done here is applied the third axiom of probability! Similarly, we can use the third axiom for calculating the probability of rolling an odd number:

In this way, all three axioms of probability make a lot of sense!