In order to get a head at least once with probability `>=0.9,`the minimum number of timesa unbiased coin needs to be tossed is

Options

A. 3
B. 4
C. 5
D. none of these

Answer

To solve the problem of determining the minimum number of times an unbiased coin needs to be tossed in order to get at least one head with a probability of at least 0.9, we can follow these steps:

### Step 1: Understand the Probability of Getting at Least One Head
The probability of getting at least one head when tossing a coin \( n \) times can be expressed as:
\[ P(\text{at least one head}) = 1 - P(\text{no heads}) \]
Where \( P(\text{no heads}) \) is the probability of getting tails in all \( n \) tosses.

### Step 2: Calculate the Probability of Getting No Heads
For an unbiased coin, the probability of getting tails in a single toss is \( \frac{1}{2} \). Therefore, the probability of getting tails in \( n \) tosses (which means no heads) is:
\[ P(\text{no heads}) = \left(\frac{1}{2}\right)^n \]

### Step 3: Set Up the Inequality
We want the probability of getting at least one head to be greater than or equal to 0.9:
\[ 1 - P(\text{no heads}) \geq 0.9 \]
Substituting the expression for \( P(\text{no heads}) \):
\[ 1 - \left(\frac{1}{2}\right)^n \geq 0.9 \]

### Step 4: Rearrange the Inequality
Rearranging the inequality gives us:
\[ \left(\frac{1}{2}\right)^n \leq 0.1 \]

### Step 5: Solve for \( n \)
To find \( n \), we can take the logarithm of both sides. Using base 2 logarithm:
\[ n \log_2\left(\frac{1}{2}\right) \leq \log_2(0.1) \]
Since \( \log_2\left(\frac{1}{2}\right) = -1 \), we can rewrite the inequality:
\[ -n \leq \log_2(0.1) \]
Thus,
\[ n \geq -\log_2(0.1) \]

### Step 6: Calculate \( \log_2(0.1) \)
Using the change of base formula:
\[ \log_2(0.1) = \frac{\log_{10}(0.1)}{\log_{10}(2)} = \frac{-1}{\log_{10}(2)} \approx \frac{-1}{0.301} \approx -3.32 \]
So,
\[ n \geq 3.32 \]

### Step 7: Determine the Minimum Integer Value for \( n \)
Since \( n \) must be a whole number, we round up to the next integer:
\[ n \geq 4 \]

### Conclusion
The minimum number of times an unbiased coin needs to be tossed to achieve a probability of at least 0.9 of getting at least one head is \( n = 4 \).