Probability tree diagram for drawing two discs from a bag with 4 purple, 5 white, and 2 black discs.

Here is the step-by-step solution to complete the tree diagram and answer the questions.

Step 1: Understand the Setup

First, let's count the total number of discs in the bag:
* Purple (P): 4
* White (W): 5
* Black (B): 2
* Total: $4 + 5 + 2 = 11$ discs.

Since the discs are not replaced, the total number of discs will drop to 10 for the second draw. The number of discs of the color drawn first will also drop by 1.

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Step 2: Complete the Tree Diagram Probabilities

First Draw Branches (Total = 11):
* Probability of Purple ($P$): $\frac{4}{11}$
* Probability of White ($W$): $\frac{5}{11}$
* Probability of Black ($B$): $\frac{2}{11}$

Second Draw Branches (Total = 10):

1. If the first was Purple (P):
* Remaining: 3 P, 5 W, 2 B.
* $P(P) = \frac{3}{10}$
* $P(W) = \frac{5}{10}$
* $P(B) = \frac{2}{10}$

2. If the first was White (W):
* Remaining: 4 P, 4 W, 2 B.
* $P(P) = \frac{4}{10}$
* $P(W) = \frac{4}{10}$
* $P(B) = \frac{2}{10}$

3. If the first was Black (B):
* Remaining: 4 P, 5 W, 1 B.
* $P(P) = \frac{4}{10}$
* $P(W) = \frac{5}{10}$
* $P(B) = \frac{1}{10}$

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Step 3: Calculate Final Probabilities for Each Path

To find the probability of a specific path, multiply the fractions along the branches.

* Top Group (Start with P):
* $P, P = \frac{4}{11} \times \frac{3}{10} = \frac{12}{110}$
* $P, W = \frac{4}{11} \times \frac{5}{10} = \frac{20}{110}$
* $P, B = \frac{4}{11} \times \frac{2}{10} = \frac{8}{110}$

* Middle Group (Start with W):
* $W, P = \frac{5}{11} \times \frac{4}{10} = \frac{20}{110}$
* $W, W = \frac{5}{11} \times \frac{4}{10} = \frac{20}{110}$
* $W, B = \frac{5}{11} \times \frac{2}{10} = \frac{10}{110}$

* Bottom Group (Start with B):
* $B, P = \frac{2}{11} \times \frac{4}{10} = \frac{8}{110}$
* $B, W = \frac{2}{11} \times \frac{5}{10} = \frac{10}{110}$
* $B, B = \frac{2}{11} \times \frac{1}{10} = \frac{2}{110}$

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Step 4: Solve Questions a(i), a(ii), and a(iii)

a. i. One purple one white
This can happen in two ways: Purple then White ($P,W$) OR White then Purple ($W,P$). We add these probabilities together.
$$ \frac{20}{110} + \frac{20}{110} = \frac{40}{110} $$
Simplifying the fraction (divide top and bottom by 10):
$$ \frac{4}{11} $$

a. ii. One black one white
This can happen in two ways: Black then White ($B,W$) OR White then Black ($W,B$).
$$ \frac{10}{110} + \frac{10}{110} = \frac{20}{110} $$
Simplifying the fraction:
$$ \frac{2}{11} $$

a. iii. The same color
This means getting two Purples ($P,P$), two Whites ($W,W$), or two Blacks ($B,B$). We add these three probabilities.
$$ \frac{12}{110} (PP) + \frac{20}{110} (WW) + \frac{2}{110} (BB) = \frac{34}{110} $$
Simplifying the fraction (divide top and bottom by 2):
$$ \frac{17}{55} $$

Final Answer:
Tree Diagram Values:
* First Draw Probabilities: P = 4/11, W = 5/11, B = 2/11
* Second Draw Probabilities (from P): 3/10, 5/10, 2/10
* Second Draw Probabilities (from W): 4/10, 4/10, 2/10
* Second Draw Probabilities (from B): 4/10, 5/10, 1/10
* Final Path Results: 12/110, 20/110, 8/110, 20/110, 20/110, 10/110, 8/110, 10/110, 2/110

Question Answers:
i. 4/11
ii. 2/11
iii. 17/55