To solve this problem, we need to calculate the probability of correctly guessing a specific 3-digit code that uses the digits 1 through 9 and is known to be odd. Let's break down the steps:
1. Total Possibilities for the Code:
- Each digit in the 3-digit code can be any number from 1 to 9.
- There are 9 choices for each of the 3 digits.
- Therefore, the total number of possible 3-digit codes with digits 1-9 is calculated by multiplying the choices for each digit together:
[tex]\[
9 \times 9 \times 9 = 729
\][/tex]
2. Conditions for the Code to be Odd:
- For a code to be odd, its last digit must be an odd number.
- The odd digits between 1 and 9 are 1, 3, 5, 7, and 9. Thus, there are 5 choices for the last digit.
- The first and second digits can still be any of the 9 possible numbers.
- Therefore, the number of possible odd 3-digit codes is:
[tex]\[
9 \times 9 \times 5 = 405
\][/tex]
3. Calculate the Probability:
- Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.
- In this case, the favorable outcome is the number of possible odd 3-digit codes, which is 405.
- The total number of possible 3-digit codes, regardless of whether they are odd or not, is 729.
- Therefore, the probability of guessing the odd code in one try is:
[tex]\[
\frac{405}{729}
\][/tex]
- Simplifying this fraction (both numbers are divisible by 81):
[tex]\[
\frac{405 \div 81}{729 \div 81} = \frac{5}{9}
\][/tex]
- Converting this fraction to a decimal:
[tex]\[
\frac{5}{9} \approx 0.555 \ldots
\][/tex]
4. Concluding the Calculation:
- The above calculation provides a good accuracy, but it can be simplified to:
[tex]\[
\frac{1}{405} \approx 0.002469
\][/tex]
Thus, the probability of correctly guessing the 3-digit code that uses the digits 1-9 and is known to be odd in one try is approximately 0.002469, or in fraction format, [tex]\(\frac{1}{405}\)[/tex].
