The number of times a die must be tossed to obtain a 6 at least one with probability exceeding `0.9` is at least

Options

A. 13
B. 19
C. 25
D. none of these

Answer

To solve the problem of determining the minimum number of times a die must be tossed to obtain at least one '6' with a probability exceeding 0.9, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the smallest integer \( n \) such that the probability of getting at least one '6' when tossing a die \( n \) times is greater than 0.9.

2. **Define the Probability**: The probability of getting at least one '6' in \( n \) tosses can be calculated using the complement rule:
\[ P(\text{at least one '6'}) = 1 - P(\text{no '6' in } n \text{ tosses}) \]

3. **Calculate the Probability of Not Getting a '6'**: The probability of not getting a '6' in a single toss of a die is:
\[ P(\text{not '6'}) = \frac{5}{6} \]
Therefore, the probability of not getting a '6' in \( n \) tosses is:
\[ P(\text{no '6' in } n \text{ tosses}) = \left(\frac{5}{6}\right)^n \]

4. **Set Up the Inequality**: We want the probability of getting at least one '6' to exceed 0.9:
\[ 1 - \left(\frac{5}{6}\right)^n > 0.9 \]

5. **Rearrange the Inequality**: This can be rearranged to:
\[ \left(\frac{5}{6}\right)^n < 0.1 \]

6. **Solve for \( n \)**: To find the smallest \( n \) that satisfies this inequality, we can take the logarithm of both sides:
\[ n \log\left(\frac{5}{6}\right) < \log(0.1) \]
Since \( \log\left(\frac{5}{6}\right) \) is negative, we can multiply both sides by -1 (which reverses the inequality):
\[ n > \frac{\log(0.1)}{\log\left(\frac{5}{6}\right)} \]

7. **Calculate the Values**: Using a calculator:
\[ \log(0.1) = -1 \]
And,
\[ \log\left(\frac{5}{6}\right) \approx -0.07918 \]
Thus,
\[ n > \frac{-1}{-0.07918} \approx 12.63 \]

8. **Determine the Smallest Integer**: Since \( n \) must be an integer, we round up to the nearest whole number:
\[ n = 13 \]

### Conclusion:
The minimum number of times a die must be tossed to obtain at least one '6' with a probability exceeding 0.9 is **13**.