To calculate the probability of opening the lock immediately by dialing a 4-digit combination code with no repeating digits, we can break it down step by step:
1. Determine the total number of ways to select 4 digits out of 10 without repetition:
- When selecting the first digit, there are 10 options (0-9).
- After selecting the first digit, there are 9 remaining options for the second digit.
- Similarly, there are 8 options for the third digit and 7 options for the fourth digit.
- Therefore, the total number of ways to select 4 digits without repetition is .
2. Determine the number of ways to select the correct 4-digit combination code without repetition:
- Since the digits cannot repeat, there is only one correct way to select the 4 digits for the combination code.
- Therefore, the number of ways to select the correct combination is 1.
3. Calculate the probability of opening the lock immediately:
- Probability =
- In this case, the favorable outcome is opening the lock with the correct combination immediately, which is 1 way.
- The total number of possible outcomes is 5040 ways.
- So, the probability of opening the lock immediately is
or approximately 0.000198 or 0.0198%.
Therefore, the probability of opening the lock immediately by dialing a 4-digit combination code with no repeating digits is approximately 0.0198%.
