This worksheet helps students visualize probability outcomes using tree diagrams for coin flips and dice rolls.

Let’s solve this step by step.

---

Problem 1: Coin thrown twice

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

The tree diagram starts like this:

- First throw:
- H (probability ½)
- Second throw: H → Outcome: H, H → Probability: ½ × ½ = ¼
- Second throw: T → Outcome: H, T → Probability: ½ × ½ = ¼
- T (probability ½)
- Second throw: H → Outcome: T, H → Probability: ½ × ½ = ¼
- Second throw: T → Outcome: T, T → Probability: ½ × ½ = ¼

So we fill in the blanks:

- After H then T → Outcome: H, T, Probability: ¼
- After T then H → Outcome: T, H, Probability: ¼
- After T then T → Outcome: T, T, Probability: ¼

Now use the tree to find:

a. P(two tails) → That’s only one outcome: T, T → Probability = ¼

b. P(one tail and one head) → This can happen in two ways: H,T or T,H → So add their probabilities: ¼ + ¼ = ½

c. P(no tails) → That means both are heads: H,H → Probability = ¼

---

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

A standard die has 6 sides: numbers 1 to 6.

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

Tree diagram:

First throw:
- Six (prob 1/6)
- Second throw: Six → Outcome: Six, Six → Prob: 1/6 × 1/6 = 1/36
- Second throw: Not Six → Outcome: Six, Not Six → Prob: 1/6 × 5/6 = 5/36
- Not Six (prob 5/6)
- Second throw: Six → Outcome: Not Six, Six → Prob: 5/6 × 1/6 = 5/36
- Second throw: Not Six → Outcome: Not Six, Not Six → Prob: 5/6 × 5/6 = 25/36

Fill in the table:

Top row (Six → Six): Outcome = Six, Six, Probability = 1/36
Next (Six → Not Six): Outcome = Six, Not Six, Probability = 5/36
Next (Not Six → Six): Outcome = Not Six, Six, Probability = 5/36
Bottom (Not Six → Not Six): Outcome = Not Six, Not Six, Probability = 25/36

Now find:

a. Two sixes → Only “Six, Six” → Probability = 1/36

b. One six → This happens in two cases: “Six, Not Six” OR “Not Six, Six” → Add them: 5/36 + 5/36 = 10/36 → Simplify: 5/18

c. At least one six → That means: either one six OR two sixes → So add:
“Six, Six” + “Six, Not Six” + “Not Six, Six” = 1/36 + 5/36 + 5/36 = 11/36

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

---

Final Answer:

Problem 1:
a. P(two tails) = 1/4
b. P(one tail and one head) = 1/2
c. P(no tails) = 1/4

Problem 2:
a. Two sixes = 1/36
b. One six = 5/18
c. At least one six = 11/36