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?