## 2.1 Undestanding terminology

- More complex events can be determined as union, intersection or complement of the simpler events

\[\begin{align} P(A \cup B)&=\text{probabillity of occurrence of A or B or at least one of them} \\ \\ P(A \cap B)&=\text{probabillity of occurrence of A and B at the same time} \\ \\ P(\overline{A})&=\text{probabillity of non-occurrence of A} \end{align}\]

- If \(A\) and \(B\) are mutually exclusive events, \(P(A \cap B)=0\), we can apply addition rule, and if they are independent we can apply multiplication rule:

\[\begin{align} P(A \cup B)&=P(A)+P(B)\\ \\ P(A \cap B)&=P(A) \cdot P(B)\\ \tag{2.3} \end{align}\]

Probability \(P(A \cap B)\) is often called joint probability

If events are not independent the conditional probability makes sense

The conditional probability \(P(A | B)\) is the probability that event \(A\) will occur given that event \(B\) has already occurred

\[\begin{align} P(A | B)&=\frac{P(A \cap B)}{P(B)}\\ \\ P(B | A)&=\frac{P(A \cap B)}{P(A)}\\ \tag{2.4} \end{align}\]

**Example 2.1 **The experiment consists of tossing two coins. Let event \(A=\)“at least one head occurred”, \(B=\)“two tails occurred”, and \(C=\)“the same outcome on both coins”. Determine the following:

\(S=\){}

\(P(A)=\)___________

\(P(B)=\)___________

\(P(C)=\)___________

Which events are mutually exclusive?

Which events are complementary?

\(P(A \cup B)=\)___________

\(P(B \cup C)=\)___________

\(P(A \cap C)=\)___________

Are events \(A\) and \(C\) independent?

\(P(A | C)=\)___________

\(P(C| A)=\)___________

**Example 2.2 **Let event \(G=\)“taking a meth class”. Let event \(H=\)“taking a science class”. Then, \(G \cap H=\)“taking a math class and a science class”. Suppose \(P(A)=0.6\), \(P(H)=0.5\), and \(P(G \cap H)=0.3\). Are \(G\) and \(H\) independent?

**Example 2.3 **One high school has 200 seniors of whom 140 will be going to college next year. Forty will be going directly to work. The remainder are taking a gap year. Fifty of the seniors going to college play sports. Thirty of the seniors going directly to work play sports. Five of the seniors taking a gap year play sports.

What is the probability that a senior is taking a gap year?

What is the probability that a senior does not play sports?

What is the probability that a senior is taking a gap year if he plays sports?