## 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:

1. $$S=$${}

2. $$P(A)=$$___________

3. $$P(B)=$$___________

4. $$P(C)=$$___________

5. Which events are mutually exclusive?

6. Which events are complementary?

7. $$P(A \cup B)=$$___________

8. $$P(B \cup C)=$$___________

9. $$P(A \cap C)=$$___________

10. Are events $$A$$ and $$C$$ independent?

11. $$P(A | C)=$$___________

12. $$P(C| A)=$$___________

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.

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

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

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