A coin is tossed n times. The probability of getting head atleast once is greater than 0.8. Then the least value of such n is

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem, we need to find the least value of \( n \) such that the probability of getting at least one head when a coin is tossed \( n \) times is greater than 0.8.

### Step-by-Step Solution:

1. **Understanding the Probability of Getting at Least One Head**:
The probability of getting at least one head when tossing a coin \( n \) times can be calculated using the complement rule. The complement of getting at least one head is getting all tails. Therefore, we can express this as:
\[ P(\text{at least one head}) = 1 - P(\text{all tails}) \]

2. **Calculating the Probability of All Tails**:
When a coin is tossed \( n \) times, the probability of getting tails in each toss is \( \frac{1}{2} \). Thus, the probability of getting all tails in \( n \) tosses is:
\[ P(\text{all tails}) = \left(\frac{1}{2}\right)^n \]

3. **Setting Up the Inequality**:
We want the probability of getting at least one head to be greater than 0.8:
\[ 1 - \left(\frac{1}{2}\right)^n > 0.8 \]

4. **Rearranging the Inequality**:
Rearranging the inequality gives:
\[ \left(\frac{1}{2}\right)^n < 0.2 \]

5. **Expressing 0.2 as a Fraction**:
We can express 0.2 as:
\[ 0.2 = \frac{1}{5} \]
So the inequality becomes:
\[ \left(\frac{1}{2}\right)^n < \frac{1}{5} \]

6. **Finding the Least Value of \( n \)**:
To find the least value of \( n \), we can test integer values of \( n \):
- For \( n = 1 \):
\[ \left(\frac{1}{2}\right)^1 = \frac{1}{2} \quad (\text{not less than } \frac{1}{5}) \]
- For \( n = 2 \):
\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \quad (\text{not less than } \frac{1}{5}) \]
- For \( n = 3 \):
\[ \left(\frac{1}{2}\right)^3 = \frac{1}{8} \quad (\text{is less than } \frac{1}{5}) \]

Therefore, the least value of \( n \) that satisfies the condition is \( n = 3 \).

### Final Answer:
The least value of \( n \) such that the probability of getting at least one head is greater than 0.8 is \( n = 3 \).