Probability of Typing Exactly 10 Letters in 12 Key Presses

This is a problem that often appears in statistical tests and is a classic example of a combinatorial probability problem. It's not uncommon to encounter such problems in high school mathematics or competitive exams. Here, we will solve the problem step by step.

Understanding the Problem

Simon was typing on a keyboard that contains 25 letters, 10 numbers, and 5 special signs. Given that he pressed a random key 12 times, what is the probability that he typed exactly 10 letters out of those 12 presses? To clarify, a single key press can result in a letter, number, or sign, and he can type a single character multiple times.

Defining the Variables

Let's first define the variables and the possible outcomes:

Total keys on the keyboard: 25 (letters) 10 (digits) 5 (signs) 40 Number of letter keys: 25 Number of non-letter keys (digits signs): 15 (10 5)

Formulating the Probability

For this problem, we need to determine the probability of an event where exactly 10 out of 12 key presses result in letters, and the remaining 2 key presses do not result in letters. To find this probability, we can use the binomial distribution formula:

P(Xk) (n choose k) * p^k * (1-p)^(n-k)

Where:

n 12 (total number of key presses) k 10 (number of letter presses) p 25/40 (probability of pressing a letter) 1-p 15/40 (probability of not pressing a letter)

Step-by-Step Calculation

1. Calculate the probability of pressing a letter:

p 25/40 5/8

2. Calculate the probability of not pressing a letter:

1-p 15/40 3/8

3. Use the binomial coefficient (n choose k):

(12 choose 10) 12! / (2! * 10!) 66

4. Substitute the values into the formula:

Probability (12 choose 10) * (5/8)^10 * (3/8)^2 66 * (5/8)^10 * (3/8)^2

5. Calculate each part separately:

(5/8)^10 9765625/1073741824 (3/8)^2 9/64 (5/8)^10 * (3/8)^2 9765625/1073741824 * 9/64 87890625/68719476736

6. Multiply the binomial coefficient by the probabilities:

Probability 66 * (87890625/68719476736) 5836537500/68719476736

7. Simplify the fraction to get the final answer:

5836537500/68719476736 ≈ 0.084412

Conclusion

Thus, the probability that Simon typed exactly 10 letters in 12 key presses, given that he pressed a random key each time, is approximately 0.084412 or 8.44%.

Additional Tips

To solve similar problems, you can follow these steps:

Define the problem and identify the variables involved. Use the appropriate binomial distribution formula. Calculate the binomial coefficient. Substitute the values into the formula and simplify. Ensure your answer is in a form that is easy to understand and interpret.

Keyword Highlights:

Probability: The likelihood of an event occurring. Keyboard typing: The act of entering text or commands using a keyboard. Combinatorics: The branch of mathematics concerning the study of finite or countable discrete structures.