Among 10000 digits,in how many cases do we expect that the digit 3 appears at most 950 times? 

To solve this problem, we can use the binomial distribution and the normal approximation to the binomial distribution. 

Given:
- Total number of digits, \( n = 10,000 \)
- Probability of a digit being 3, \( p = \frac{1}{10} = 0.1 \)
- We are interested in the number of cases where the digit 3 appears at most 950 times.

First, let's calculate the mean and the standard deviation of the binomial distribution:
\[ \mu = np = 10,000 \times 0.1 = 1,000 \]
\[ \sigma = \sqrt{np(1-p)} = \sqrt{10,000 \times 0.1 \times 0.9} = \sqrt{900} = 30 \]

We want to find the probability that the number of times digit 3 appears, \( X \), is at most 950. Using the normal approximation:
\[ Z = \frac{X - \mu}{\sigma} \]

For \( X = 950 \):
\[ Z = \frac{950 - 1,000}{30} = \frac{-50}{30} \approx -1.67 \]

Using the standard normal distribution table, we can find the probability corresponding to \( Z = -1.67 \):
\[ P(Z \leq -1.67) \approx 0.0475 \]

Therefore, the probability that the digit 3 appears at most 950 times out of 10,000 digits is approximately 4.75%.

To find the expected number of such cases among 10,000 trials:
\[ \text{Expected number of cases} = 10,000 \times 0.0475 = 475 \]

So, we expect that in 475 cases out of 10,000, the digit 3 will appear at most 950 times.

Would you like any more details or have any other questions?

Here are 5 questions you might consider next:
1. How do we use the normal approximation for other binomial probabilities?
2. Can you explain how the binomial distribution is derived?
3. How do we calculate probabilities for a non-symmetric distribution?
4. What are some examples of real-world applications of the binomial distribution?
5. How does the Central Limit Theorem apply to the normal approximation of binomial distributions?

**Tip:** When using the normal approximation to the binomial distribution, ensure the sample size \( n \) and probability \( p \) are such that \( np \) and \( n(1-p) \) are both greater than 5 for a more accurate approximation.

Explore the probability calculation for the appearance of the digit 3 in 10,000 digits, using binomial and normal distribution theories. Learn how to apply the central limit theorem for accurate approximations. Probability calculation explained with detailed steps and formulas.

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