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?