Step 1: Understanding the Concept:
The unit's digit of a product depends only on the unit's digits of its factors. For a product to have a unit's digit of 1, 3, 7, or 9, the product must be odd and must not be a multiple of 5. This implies that none of the factors can be even and none of the factors can end in 5.
Step 2: Key Formula or Approach:
1. Identify the unit digits of individual factors that lead to the desired product units digit.
2. Probability \( P = (\text{Prob of one factor satisfies condition})^4 \).
Step 3: Detailed Explanation:
Possible unit digits for a natural number are \( \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \). Total = 10 digits.
Condition for the product to end in 1, 3, 7, or 9:
- The product is not divisible by 2 (all factors must be odd).
- The product is not divisible by 5 (none of the factors can end in 5).
So, the allowed unit digits for each factor are \( \{1, 3, 7, 9\} \).
Number of favorable unit digits for one number \( = 4 \).
Probability for one number to satisfy this condition \( = 4/10 = 2/5 \).
Since the four numbers are selected independently and multiplied:
\[ P(\text{Product ends in 1, 3, 7, 9}) = (2/5) \times (2/5) \times (2/5) \times (2/5) \]
\[ P = \frac{16}{625} \]
Step 4: Final Answer:
The probability is \( 16/625 \).