Question
For many computer tablets, the owner can set a 4-digit pass code to unlock the device. If the digits of a passcode are chosen at random without replacement from the digits 0 - 9, what is the probability that the passcode is 1234?
Answers
GPT-4o mini
To find the probability that a randomly chosen 4-digit passcode is specifically "1234", we need to first determine the total number of possible 4-digit passcodes that can be formed using the digits 0 through 9 without replacement.
1. **Finding Total Ways to Form a 4-Digit Passcode:**
- The first digit can be any of the 10 digits (0 through 9).
- The second digit can be any of the remaining 9 digits (since we cannot use the digit chosen as the first).
- The third digit can be any of the remaining 8 digits.
- The fourth digit can be any of the remaining 7 digits.
Thus, the total number of different 4-digit passcodes can be calculated as:
\[
10 \times 9 \times 8 \times 7
\]
Calculating this gives:
\[
10 \times 9 = 90,
\]
\[
90 \times 8 = 720,
\]
\[
720 \times 7 = 5040.
\]
Therefore, there are \( 5040 \) possible 4-digit passcodes that could be formed from the digits 0 to 9 without replacement.
2. **Finding the Probability of Getting "1234":**
- There is exactly 1 specific passcode that is "1234".
3. **Calculating the Probability:**
- The probability \( P \) of randomly selecting the passcode "1234" is calculated by taking the number of successful outcomes (which is 1) over the total number of possible outcomes (which is 5040).
\[
P(\text{passcode is } 1234) = \frac{\text{Number of successful outcomes}}{\text{Total number of outcomes}} = \frac{1}{5040}.
\]
Thus, the probability that the passcode is "1234" is:
\[
\boxed{\frac{1}{5040}}.
\]
1. **Finding Total Ways to Form a 4-Digit Passcode:**
- The first digit can be any of the 10 digits (0 through 9).
- The second digit can be any of the remaining 9 digits (since we cannot use the digit chosen as the first).
- The third digit can be any of the remaining 8 digits.
- The fourth digit can be any of the remaining 7 digits.
Thus, the total number of different 4-digit passcodes can be calculated as:
\[
10 \times 9 \times 8 \times 7
\]
Calculating this gives:
\[
10 \times 9 = 90,
\]
\[
90 \times 8 = 720,
\]
\[
720 \times 7 = 5040.
\]
Therefore, there are \( 5040 \) possible 4-digit passcodes that could be formed from the digits 0 to 9 without replacement.
2. **Finding the Probability of Getting "1234":**
- There is exactly 1 specific passcode that is "1234".
3. **Calculating the Probability:**
- The probability \( P \) of randomly selecting the passcode "1234" is calculated by taking the number of successful outcomes (which is 1) over the total number of possible outcomes (which is 5040).
\[
P(\text{passcode is } 1234) = \frac{\text{Number of successful outcomes}}{\text{Total number of outcomes}} = \frac{1}{5040}.
\]
Thus, the probability that the passcode is "1234" is:
\[
\boxed{\frac{1}{5040}}.
\]