Three coins are tossed once. Find the probability of getting at most two heads.

Answer

To solve the problem of finding the probability of getting at most two heads when three coins are tossed, we can follow these steps:

### Step 1: Identify the Sample Space
When three coins are tossed, each coin has two possible outcomes: heads (H) or tails (T). Therefore, the total number of outcomes when tossing three coins can be calculated as follows:

\[ \text{Total outcomes} = 2^n \]
where \( n \) is the number of coins. Here, \( n = 3 \).

\[ \text{Total outcomes} = 2^3 = 8 \]

The sample space (S) consists of the following outcomes:
- HHH
- HHT
- HTH
- THH
- HTT
- THT
- TTH
- TTT

### Step 2: Define the Favorable Outcomes
Next, we need to find the outcomes that correspond to getting at most two heads. "At most two heads" means we can have 0, 1, or 2 heads.

- **0 heads**: TTT
- **1 head**: HTT, THT, TTH (3 outcomes)
- **2 heads**: HHT, HTH, THH (3 outcomes)

Now, let's list all the favorable outcomes:
- 0 heads: TTT
- 1 head: HTT, THT, TTH
- 2 heads: HHT, HTH, THH

Counting these, we have:
- 1 outcome with 0 heads
- 3 outcomes with 1 head
- 3 outcomes with 2 heads

Thus, the total number of favorable outcomes is:
\[ 1 + 3 + 3 = 7 \]

### Step 3: Calculate the Probability
The probability (P) of an event is given by the formula:
\[ P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

Substituting the values we found:
\[ P(\text{At most 2 heads}) = \frac{7}{8} \]

### Final Answer
Thus, the probability of getting at most two heads when three coins are tossed is:
\[ \frac{7}{8} \]

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