The type of probability calculations you've been dealing with fall under combinatorial probability and binomial probability.

Key Concepts:

1. Combinatorial Probability: This involves counting the number of ways certain events can occur and then determining the probability based on these counts. For example, when calculating how many ways the digit 5 can appear in a 5-digit number, we use combinations to figure out the different possibilities (like using $\binom{n}{k}$ for choosing positions).

2. Binomial Probability: This is used when you are dealing with a fixed number of independent trials, each with two possible outcomes (like a digit being either 5 or not 5). The binomial distribution formula is given by: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$ Where:

   • $n$ is the number of trials (e.g., 5-digit number has 5 trials),
   • $k$ is the number of successes (e.g., how many times the digit 5 appears),
   • $p$ is the probability of success on each trial (e.g., $\frac{1}{10}$ for the digit 5),
   • $1-p$ is the probability of failure.

Useful Resources for Learning:

• Textbooks on Probability and Statistics: Look for sections on combinatorial probability and the binomial distribution.
• Online Courses: Websites like Khan Academy, Coursera, and edX offer courses that cover these topics in detail.
• Probability Worksheets and Exercises: Practicing problems related to combinatorial and binomial probability will help solidify these concepts.
• Probability Theory Books: Books like "Introduction to Probability" by Joseph K. Blitzstein or "A First Course in Probability" by Sheldon Ross can provide a deeper understanding.

Further Questions You Could Explore:

1. How does the Poisson distribution relate to the binomial distribution, especially when considering rare events?
2. Can combinatorial probability be applied to continuous distributions, or is it limited to discrete scenarios?
3. What is the difference between permutations and combinations, and how do they affect probability calculations?
4. How do conditional probabilities play a role in scenarios where multiple events are dependent?
5. What are the applications of these probability principles in real-world scenarios, like genetics or quality control?

Tip:

When learning combinatorial and binomial probability, practice with different scenarios to understand how the formulas adapt to various contexts. Visual aids, like tree diagrams or probability tables, can also be helpful in breaking down complex problems.