Three coins are tossed together. List the sample space for the event. Find the probability of atleast two heads.

Answer

To solve the problem step by step, we will first list the sample space when three coins are tossed and then calculate the probability of getting at least two heads.

### Step 1: List the Sample Space
When three coins are tossed, each coin can either show Heads (H) or Tails (T). The total number of outcomes when tossing three coins can be calculated using the formula \(2^n\), where \(n\) is the number of coins.

**Calculation:**
- Number of coins, \(n = 3\)
- Total outcomes = \(2^3 = 8\)

Now, we can list all possible outcomes:
1. HHH (3 heads)
2. HHT (2 heads, 1 tail)
3. HTH (2 heads, 1 tail)
4. HTT (1 head, 2 tails)
5. THH (2 heads, 1 tail)
6. THT (1 head, 2 tails)
7. TTH (1 head, 2 tails)
8. TTT (3 tails)

**Sample Space:**
\[ S = \{ \text{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} \} \]

### Step 2: Identify Favorable Outcomes for At Least Two Heads
Next, we need to find the outcomes that have at least two heads. The favorable outcomes are:
1. HHH (3 heads)
2. HHT (2 heads, 1 tail)
3. HTH (2 heads, 1 tail)
4. THH (2 heads, 1 tail)

**Favorable Outcomes:**
\[ F = \{ \text{HHH, HHT, HTH, THH} \} \]

### Step 3: Count the Favorable Outcomes
Now, we count the number of favorable outcomes:
- Total favorable outcomes = 4 (HHH, HHT, HTH, THH)

### Step 4: Calculate the Total Outcomes
As calculated earlier, the total number of outcomes when tossing three coins is:
- Total outcomes = 8

### Step 5: Calculate the Probability
The probability of an event is given by the formula:
\[ P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} \]

**Calculation:**
\[ P(\text{at least 2 heads}) = \frac{4}{8} = \frac{1}{2} \]

### Final Answer
The probability of getting at least two heads when tossing three coins is \(\frac{1}{2}\).

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