1. **State the problem:** We want to find two probabilities based on the Venn diagram data of 45 dancers who like tango (T), hip-hop (H), or quick-step (Q). 2. **Identify the numbers from the Venn diagram:** - T only = 9 - T & H only = 2 - T & Q only = 8 - T & H & Q = 1 - H only = 7 - H & Q only = 3 - Q only = 10 - Outside all circles = 5 3. **Calculate the total number who like tango (T):** $$9 + 2 + 8 + 1 = 20$$ 4. **Find the number who like both tango and hip-hop:** This includes T & H only plus T & H & Q: $$2 + 1 = 3$$ 5. **Calculate the probability that a randomly selected tango lover also likes hip-hop:** $$P(H|T) = \frac{\text{Number liking both } T \text{ and } H}{\text{Number liking } T} = \frac{3}{20}$$ 6. **Calculate the number who do NOT like hip-hop:** Total dancers = 45 Number who like hip-hop = sum of H only, H & T only, H & Q only, and T & H & Q: $$7 + 2 + 3 + 1 = 13$$ Number who do NOT like hip-hop: $$45 - 13 = 32$$ 7. **Find the number who do NOT like hip-hop but like quick-step:** This includes Q only plus T & Q only (because they include Q and not H), excluding any overlaps with H. From the Venn diagram: - Q only = 10 - T & Q only = 8 - T & H & Q includes H, so exclude it - H & Q only includes H, so exclude it So the number who like Q but NOT H: $$10 + 8 = 18$$ 8. **Calculate the probability that a dancer selected at random from those who don't like hip-hop does like the quick-step:** $$P(Q|\text{not } H) = \frac{18}{32} = \frac{9}{16}$$ **Final answers:** - Probability a tango dancer also likes hip-hop: $\frac{3}{20}$ - Probability a non-hip-hop dancer likes quick-step: $\frac{9}{16}$