To construct a probability tree for the selection of two balls from the urn without replacement, let us proceed step by step.

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Step 1: Information about the urn

• Total balls: $6 + 7 + 8 = 21$
• Red balls ($R$): 6
• White balls ($W$): 7
• Black balls ($B$): 8

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Step 2: Create the tree for the first selection

The probabilities for the first selection of each color are:

• Probability of selecting $R$: $\frac{6}{21}$
• Probability of selecting $W$: $\frac{7}{21}$
• Probability of selecting $B$: $\frac{8}{21}$

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Step 3: Create branches for the second selection

Since it is without replacement, the probabilities for the second selection depend on what was chosen first. The total number of balls decreases by 1 after the first selection.

• If the first ball is $R$ ($6 \to 5$):

  • Probability of $R$ next: $\frac{5}{20}$
  • Probability of $W$ next: $\frac{7}{20}$
  • Probability of $B$ next: $\frac{8}{20}$

• If the first ball is $W$ ($7 \to 6$):

  • Probability of $R$ next: $\frac{6}{20}$
  • Probability of $W$ next: $\frac{6}{20}$
  • Probability of $B$ next: $\frac{8}{20}$

• If the first ball is $B$ ($8 \to 7$):

  • Probability of $R$ next: $\frac{6}{20}$
  • Probability of $W$ next: $\frac{7}{20}$
  • Probability of $B$ next: $\frac{7}{20}$

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Step 4: Sketch the tree

Here’s how the tree would look (simplified for clarity):

1. First selection:

   • $R$: $\frac{6}{21}$
   • $W$: $\frac{7}{21}$
   • $B$: $\frac{8}{21}$

2. Second selection (conditional probabilities):

   • After $R$: $R$ ($\frac{5}{20}$), $W$ ($\frac{7}{20}$), $B$ ($\frac{8}{20}$)
   • After $W$: $R$ ($\frac{6}{20}$), $W$ ($\frac{6}{20}$), $B$ ($\frac{8}{20}$)
   • After $B$: $R$ ($\frac{6}{20}$), $W$ ($\frac{7}{20}$), $B$ ($\frac{7}{20}$)

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Visual Representation of the Tree

I cannot directly draw the tree, but here's the textual structure:

1. $R (\frac{6}{21})$
   • $R (\frac{5}{20})$
   • $W (\frac{7}{20})$
   • $B (\frac{8}{20})$
2. $W (\frac{7}{21})$
   • $R (\frac{6}{20})$
   • $W (\frac{6}{20})$
   • $B (\frac{8}{20})$
3. $B (\frac{8}{21})$
   • $R (\frac{6}{20})$
   • $W (\frac{7}{20})$
   • $B (\frac{7}{20})$

Let me know if you would like me to create a graphical tree or clarify further!

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Expansion Questions:

1. How can this tree be applied to calculate the probability of selecting two red balls?
2. What is the probability of selecting one red ball and one black ball in either order?
3. How does the tree change if replacement is allowed?
4. What is the probability of selecting two balls of the same color?
5. How can this probability tree be converted into a table for analysis?

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Tip:

When working with probability trees, always verify that the sum of probabilities at each branch equals 1 to ensure accuracy!