1. **State the problem:** We have a population with values $0,0,5,7,8,9,9,10,12,14,15,16,18,19,19$ and a sample size of 14. We want to find the tree diagram of the probability of the population. 2. **Understanding the problem:** A tree diagram for probability shows all possible outcomes and their probabilities step-by-step. Here, the population has 15 values, but the sample size is 14, so we consider sampling without replacement. 3. **Calculate total population size:** The population size $N=15$. 4. **Sampling without replacement:** The probability of each sample depends on previous picks. For the first pick, probability of each value is $\frac{\text{count of that value}}{15}$. 5. **Example for first pick probabilities:** - Value 0 appears twice, so $P(0) = \frac{2}{15}$ - Value 5 appears once, so $P(5) = \frac{1}{15}$ - Value 7 appears once, so $P(7) = \frac{1}{15}$ - Value 8 appears once, so $P(8) = \frac{1}{15}$ - Value 9 appears twice, so $P(9) = \frac{2}{15}$ - Value 10 appears once, so $P(10) = \frac{1}{15}$ - Value 12 appears once, so $P(12) = \frac{1}{15}$ - Value 14 appears once, so $P(14) = \frac{1}{15}$ - Value 15 appears once, so $P(15) = \frac{1}{15}$ - Value 16 appears once, so $P(16) = \frac{1}{15}$ - Value 18 appears once, so $P(18) = \frac{1}{15}$ - Value 19 appears twice, so $P(19) = \frac{2}{15}$ 6. **Subsequent picks:** After picking one value, reduce the count of that value by 1 and total population by 1, then recalculate probabilities for the next pick. 7. **Tree diagram structure:** Each branch represents a pick with its probability. The product of probabilities along a path gives the probability of that sequence. 8. **Summary:** The tree diagram is complex due to many branches (14 picks), but the method is: - Start with initial probabilities based on counts over 15 - For each pick, update counts and total - Calculate probabilities for next pick - Multiply probabilities along branches This is the conceptual approach to build the tree diagram for the population sampling without replacement with sample size 14.