Probability tree diagram for drawing discs without replacement, showing outcomes and probabilities for combinations of purple, white, and black discs.

Let’s solve this step by step.

We are told:
- Bag has 4 purple (P), 5 white (W), and 2 black (B) discs → total = 4 + 5 + 2 = 11 discs
- We draw one disc, do not replace it, then draw a second disc.
- We need to complete the tree diagram probabilities and answer parts i, ii, iii.

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Step 1: First Draw Probabilities

Total discs at start: 11

- P(first = Purple) = 4/11
- P(first = White) = 5/11
- P(first = Black) = 2/11

These go on the first branches of the tree.

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Step 2: Second Draw Probabilities (after removing one disc)

After drawing one disc, there are now 10 discs left.

#### If first was Purple (4 taken out → 3 left):

- P(second = Purple | first = Purple) = 3/10
- P(second = White | first = Purple) = 5/10
- P(second = Black | first = Purple) = 2/10

#### If first was White (5 taken out → 4 left):

- P(second = Purple | first = White) = 4/10
- P(second = White | first = White) = 4/10
- P(second = Black | first = White) = 2/10

#### If first was Black (2 taken out → 1 left):

- P(second = Purple | first = Black) = 4/10
- P(second = White | first = Black) = 5/10
- P(second = Black | first = Black) = 1/10

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Step 3: Fill in Tree Diagram Branches (for reference — we’ll use these for calculations)

First branch:
→ P: 4/11
→ P: 3/10 → joint prob = (4/11)(3/10) = 12/110
 → W: 5/10 → joint prob = (4/11)(5/10) = 20/110
 → B: 2/10 → joint prob = (4/11)(2/10) = 8/110

→ W: 5/11
 → P: 4/10 → joint prob = (5/11)(4/10) = 20/110
 → W: 4/10 → joint prob = (5/11)(4/10) = 20/110
 → B: 2/10 → joint prob = (5/11)(2/10) = 10/110

→ B: 2/11
 → P: 4/10 → joint prob = (2/11)(4/10) = 8/110
 → W: 5/10 → joint prob = (2/11)(5/10) = 10/110
 → B: 1/10 → joint prob = (2/11)(1/10) = 2/110

Note: All joint probabilities add up to 110/110 = 1 → good check!

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Now Answer Part a:

#### i. One purple and one white

This can happen in two ways:
- First purple, then white → (4/11)(5/10) = 20/110
- First white, then purple → (5/11)(4/10) = 20/110

Add them: 20/110 + 20/110 = 40/110

Simplify: divide numerator and denominator by 10 → 4/11

Wait — let me double-check: 40 ÷ 10 = 4, 110 ÷ 10 = 11 → yes, 4/11.

But actually, 40/110 reduces further? GCD of 40 and 110 is 10 → so 4/11 is simplest form.

✔ So, 4/11

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#### ii. One black and one white

Two ways:
- First black, then white → (2/11)(5/10) = 10/110
- First white, then black → (5/11)(2/10) = 10/110

Add: 10/110 + 10/110 = 20/110

Simplify: divide by 10 → 2/11

✔ So, 2/11

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#### iii. The same color

Three cases:
- Both purple: (4/11)(3/10) = 12/110
- Both white: (5/11)(4/10) = 20/110
- Both black: (2/11)(1/10) = 2/110

Add: 12 + 20 + 2 = 32 → 32/110

Simplify: divide numerator and denominator by 2 → 16/55

Check: 32 ÷ 2 = 16, 110 ÷ 2 = 55 → yes.

✔ So, 16/55

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Final Answers:

i. One purple one white → 4/11
ii. One black one white → 2/11
iii. Same color → 16/55

All checked and verified.

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Final Answer:
a. i. \(\frac{4}{11}\)
 ii. \(\frac{2}{11}\)
iii. \(\frac{16}{55}\)