A coin is tossed successively three times. The probability of getting exactly one head or 2 heads, is

Options

A. `1/4`
B. `1/2`
C. `3/4`
D. `1/3`

Answer

To solve the problem of finding the probability of getting exactly one head or exactly two heads when a coin is tossed three times, we can follow these steps:

### Step 1: Identify the Sample Space
When a coin is tossed three times, the sample space consists of all possible outcomes. The outcomes can be represented as:
- HHH (3 heads)
- HHT (2 heads, 1 tail)
- HTH (2 heads, 1 tail)
- THH (2 heads, 1 tail)
- HTT (1 head, 2 tails)
- THT (1 head, 2 tails)
- TTH (1 head, 2 tails)
- TTT (0 heads)

Thus, the total sample space is:
\[ S = \{ HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \} \]

### Step 2: Count the Total Outcomes
The total number of outcomes when a coin is tossed three times is:
\[ \text{Total Outcomes} = 2^3 = 8 \]

### Step 3: Identify Favorable Outcomes
Next, we need to find the favorable outcomes for getting exactly one head and exactly two heads.

**Exactly One Head:**
The outcomes with exactly one head are:
- HTT
- THT
- TTH

So, the number of outcomes with exactly one head is:
\[ \text{Favorable Outcomes (1 head)} = 3 \]

**Exactly Two Heads:**
The outcomes with exactly two heads are:
- HHT
- HTH
- THH

So, the number of outcomes with exactly two heads is:
\[ \text{Favorable Outcomes (2 heads)} = 3 \]

### Step 4: Calculate Total Favorable Outcomes
The total favorable outcomes for getting exactly one head or exactly two heads is:
\[ \text{Total Favorable Outcomes} = 3 + 3 = 6 \]

### Step 5: Calculate the Probability
The probability of getting exactly one head or exactly two heads is given by the formula:
\[ P(\text{exactly 1 head or 2 heads}) = \frac{\text{Total Favorable Outcomes}}{\text{Total Outcomes}} \]

Substituting the values:
\[ P = \frac{6}{8} = \frac{3}{4} \]

### Final Answer
The probability of getting exactly one head or exactly two heads when a coin is tossed three times is:
\[ \frac{3}{4} \]

---