1. **Problem Statement:** Given the table of students by gender and major, find various probabilities related to gender and major. 2. **Given Table:** | Major | Accounting | Management | Finance | Total | |-----------|------------|------------|---------|-------| | Male | 100 | 150 | 50 | 300 | | Female | 100 | 50 | 50 | 200 | | Total | 200 | 200 | 100 | 500 | 3. **Formulas and Rules:** - Probability of event $A$: $P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$ - Addition rule for union of two events $A$ and $B$: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ - Conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ - Independence: $A$ and $B$ are independent if $P(A \cap B) = P(A)P(B)$ 4. **Calculations:** **a. Probability of selecting a female student:** $$P(\text{Female}) = \frac{200}{500} = 0.4$$ **b. Probability of selecting a finance or accounting major:** $$P(\text{Finance or Accounting}) = \frac{100 + 200}{500} = \frac{300}{500} = 0.6$$ **c. Probability of selecting a female or an accounting major:** - $P(\text{Female}) = \frac{200}{500} = 0.4$ - $P(\text{Accounting}) = \frac{200}{500} = 0.4$ - $P(\text{Female} \cap \text{Accounting}) = \frac{100}{500} = 0.2$ Using addition rule: $$P(\text{Female} \cup \text{Accounting}) = P(\text{Female}) + P(\text{Accounting}) - P(\text{Female} \cap \text{Accounting}) = 0.4 + 0.4 - 0.2 = 0.6$$ **Rule applied:** General addition rule for union of two events. **d. Are gender and major independent?** - Check if $P(\text{Female} \cap \text{Accounting}) = P(\text{Female})P(\text{Accounting})$ - $P(\text{Female} \cap \text{Accounting}) = 0.2$ - $P(\text{Female})P(\text{Accounting}) = 0.4 \times 0.4 = 0.16$ Since $0.2 \neq 0.16$, gender and major are **not independent**. **e. Probability of selecting an accounting major given the person is male:** $$P(\text{Accounting} | \text{Male}) = \frac{P(\text{Accounting} \cap \text{Male})}{P(\text{Male})} = \frac{100/500}{300/500} = \frac{100}{300} = \frac{1}{3} \approx 0.3333$$ **f. Probability that both selected students are accounting majors:** - Total students: 500 - Accounting majors: 200 Probability first selected is accounting major: $$\frac{200}{500} = 0.4$$ Probability second selected is accounting major (without replacement): $$\frac{199}{499} \approx 0.3988$$ Combined probability: $$0.4 \times 0.3988 = 0.1595$$ **Final answers:** - a. $0.4$ - b. $0.6$ - c. $0.6$ (using addition rule) - d. No, not independent - e. $\frac{1}{3}$ - f. $0.1595$