Worksheet on Tree Diagrams (A) for probability calculations, featuring coin toss and dice throw examples.

Let’s solve this step by step.

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Problem 1: Coin thrown twice

We’re throwing a coin two times. Each throw has two possible outcomes: Heads (H) or Tails (T), each with probability ½.

The tree diagram starts like this:

- First throw: H (prob ½) → then second throw: H or T
- First throw: T (prob ½) → then second throw: H or T

So the four possible outcomes are:

1. H, H → already given: prob = ½ × ½ = ¼
2. H, T → outcome is “H, T”, prob = ½ × ½ = ¼
3. T, H → outcome is “T, H”, prob = ½ × ½ = ¼
4. T, T → outcome is “T, T”, prob = ½ × ½ = ¼

Now fill in the blanks in the table:

- Second row (after H → T): Outcome = H, T, Probability = ¼
- Third row (after T → H): Outcome = T, H, Probability = ¼
- Fourth row (after T → T): Outcome = T, T, Probability = ¼

Now use the tree to answer:

a. P(two tails) → that’s T, T → ¼

b. P(one tail and one head) → this can happen two ways: H,T or T,H → so ¼ + ¼ = ½

c. P(no tails) → that means both heads → H,H → ¼

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Problem 2: Lilly throws a dice twice, trying for sixes

A standard die has 6 sides: 1,2,3,4,5,6.

Probability of rolling a six = 1/6
Probability of NOT rolling a six = 5/6

Tree diagram:

First throw:
- Six (prob 1/6) → then second throw: Six or Not Six
- Not Six (prob 5/6) → then second throw: Six or Not Six

Outcomes and probabilities:

1. Six, Six → prob = 1/6 × 1/6 = 1/36
2. Six, Not Six → prob = 1/6 × 5/6 = 5/36
3. Not Six, Six → prob = 5/6 × 1/6 = 5/36
4. Not Six, Not Six → prob = 5/6 × 5/6 = 25/36

Fill in the table:

Top branch (Six → ?):
- First box: Six → Outcome: Six, Six, Prob: 1/36
- Second box: Not Six → Outcome: Six, Not Six, Prob: 5/36

Bottom branch (Not Six → ?):
- Third box: Six → Outcome: Not Six, Six, Prob: 5/36
- Fourth box: Not Six → Outcome: Not Six, Not Six, Prob: 25/36

Now find:

a. Two sixes → only “Six, Six” → 1/36

b. One six → this happens in two cases: “Six, Not Six” OR “Not Six, Six” → 5/36 + 5/36 = 10/36 → simplify to 5/18

c. At least one six → this includes:
- Two sixes (1/36)
- One six (10/36)
Total = 1/36 + 10/36 = 11/36

(Alternatively: 1 – P(no sixes) = 1 – 25/36 = 11/36 — same answer!)

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Final Answer:

Problem 1:

Completed tree diagram outcomes and probabilities:

- H, T → ¼
- T, H → ¼
- T, T → ¼

a. P(two tails) = ¼
b. P(one tail and one head) = ½
c. P(no tails) = ¼

Problem 2:

Completed tree diagram:

- Six, Six → 1/36
- Six, Not Six → 5/36
- Not Six, Six → 5/36
- Not Six, Not Six → 25/36

a. Two sixes = 1/36
b. One six = 5/18
c. At least one six = 11/36