Step by Step Free Solution From UpStudy:

To determine the probability that a four-digit code, which uses the digits 0−9 without repeating any digits, only contains odd numbers, we can follow these steps:

1. Count the total number of possible four-digit codes:   

- There are 10 possible digits (0 through 9).   

- Since the digits cannot be repeated, we have:     

- 10 choices for the first digit,     

- 9 choices for the second digit,     

- 8 choices for the third digit,     

- 7 choices for the fourth digit.   

- Therefore, the total number of possible four-digit codes is: \[10 \times 9 \times 8 \times 7 = 5040\]  

2. Count the number of four-digit codes that only contain odd numbers:   

- There are 5 odd digits (1, 3, 5, 7, 9).   

- Since the digits cannot be repeated, we have:     

- 5 choices for the first digit,     

- 4 choices for the second digit,     

- 3 choices for the third digit,     

- 2 choices for the fourth digit.   

- Therefore, the number of four-digit codes that contain only odd numbers is: \[5 \times 4 \times 3 \times 2 = 120\]  

3. Calculate the probability:   

- The probability that the code contains only odd numbers is the number of favorable outcomes divided by the total number of possible outcomes:     \[\frac { 120} { 5040} = \frac { 1} { 42} \approx 0.02381\]  

Therefore, the approximate probability that the code only contains odd numbers is: \[0.02381\]

Key Concepts:

1. Total Number of Possible Outcomes: When calculating probabilities, it's essential to determine the total number of possible outcomes. For a four-digit code using digits 0-9 without repetition, this is calculated by multiplying the number of choices for each digit.

2. Favorable Outcomes: These are the specific outcomes that satisfy the condition of the probability problem—in this case, four-digit codes using only odd numbers.

3. Probability Calculation: The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes.

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