Question: Suppose students are doing research for an English assignment and randomly use two different search engines to find information. The two-way table below shows the search engine that the students used, and whether or not they earned a passing score on their assignment. Pass Fail Total Search Engine A (A) 64 16 80 Search Engine B (B) 96 24 120 Total 160 40 200 Use the data to answer the following questions. Any probability should be entered as a number between 0 and 1, inclusive. How many papers were completed? P(B) P(Pass) P(B and Pass) P(B | Pass) P(Pass | B) Are the events Pass and B independent? Yes No There is not enough information.

1. **State the problem:** We have a two-way table showing the number of students who used search engines A and B and whether they passed or failed an assignment. We want to find various probabilities and check if the events Pass and B are independent. 2. **Given data:** - Pass and A: $64$ - Fail and A: $16$ - Total A: $80$ - Pass and B: $96$ - Fail and B: $24$ - Total B: $120$ - Total Pass: $160$ - Total Fail: $40$ - Total papers: $200$ 3. **How many papers were completed?** - Total papers completed = $200$ 4. **Calculate $P(B)$:** $$P(B) = \frac{\text{Total papers using B}}{\text{Total papers}} = \frac{120}{200} = 0.6$$ 5. **Calculate $P(Pass)$:** $$P(Pass) = \frac{\text{Total passing papers}}{\text{Total papers}} = \frac{160}{200} = 0.8$$ 6. **Calculate $P(B \text{ and } Pass)$:** $$P(B \text{ and } Pass) = \frac{\text{Papers passing using B}}{\text{Total papers}} = \frac{96}{200} = 0.48$$ 7. **Calculate $P(B | Pass)$:** $$P(B | Pass) = \frac{P(B \text{ and } Pass)}{P(Pass)} = \frac{0.48}{0.8} = 0.6$$ 8. **Calculate $P(Pass | B)$:** $$P(Pass | B) = \frac{P(B \text{ and } Pass)}{P(B)} = \frac{0.48}{0.6} = 0.8$$ 9. **Check independence:** - Two events $A$ and $B$ are independent if and only if: $$P(A \text{ and } B) = P(A) \times P(B)$$ - Check if $P(Pass \text{ and } B) = P(Pass) \times P(B)$: $$0.48 \stackrel{?}{=} 0.8 \times 0.6 = 0.48$$ - Since both sides are equal, Pass and B are independent events. **Final answers:** - Total papers completed: $200$ - $P(B) = 0.6$ - $P(Pass) = 0.8$ - $P(B \text{ and } Pass) = 0.48$ - $P(B | Pass) = 0.6$ - $P(Pass | B) = 0.8$ - Are Pass and B independent? **Yes**