Question

1st toss 2nd toss 3rd toss outcomes
h
├─ h ── h −−− (h,h,h)
│ └─ t −−− (h,h,t)
└─ t ├─ h −−− (h,t,h)
└─ t −−− (h,t,t)
t
├─ h ├─ h −−− (t,h,h)
│ └─ t −−− (t,h,t)
└─ t ├─ h −−− (t,t,h)
└─ t −−− (t,t,t)
this tree diagram shows all the possible outcomes for tossing a coin three times.
how many possible outcomes are there when a coin is tossed 3 times?
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how many outcomes include exactly one tail?
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Explanation:

First Question: How many possible outcomes are there when a coin is tossed 3 times?

Step1: Analyze the tree diagram

Looking at the tree diagram, we can count the number of end - points (outcomes). Each toss of the coin has 2 possibilities (H or T). For the first toss, 2 options; for each of those, the second toss has 2 options, and for each of the second - toss options, the third toss has 2 options. But we can also directly count from the tree.
Counting the outcomes: (H, H, H), (H, H, T), (H, T, H), (H, T, T), (T, H, H), (T, H, T), (T, T, H), (T, T, T).

Step2: Count the number of outcomes

We can see that there are 8 outcomes. Mathematically, for a coin tossed \(n\) times, the number of possible outcomes is \(2^n\). Here \(n = 3\), so \(2^3=8\).

Step1: Identify outcomes with exactly one T

We go through each outcome:

• (H, H, T): has 1 tail.
• (H, T, H): has 1 tail.
• (T, H, H): has 1 tail.

The other outcomes either have 0 tails (e.g., (H, H, H)) or 2 or 3 tails (e.g., (H, T, T), (T, H, T), (T, T, H), (T, T, T)).

Step2: Count these outcomes

We can see that there are 3 such outcomes.

Answer:

8

Second Question: How many outcomes include exactly one tail?