To find the probability that the last digit of the product of four selected numbers is 1, 3, 5, or 7, we need to consider the possible combinations of the last digits of the four numbers.

Let's analyze the last digit possibilities for each number:

1. The last digit of any number multiplied by 1 will be the same as the last digit of that number.
2. The last digit of any number multiplied by 2 will be even.
3. The last digit of any number multiplied by 3 will be odd.
4. The last digit of any number multiplied by 4 will be even.
5. The last digit of any number multiplied by 5 will end in 0 or 5.
6. The last digit of any number multiplied by 6 will be even.
7. The last digit of any number multiplied by 7 will be odd.
8. The last digit of any number multiplied by 8 will be even.
9. The last digit of any number multiplied by 9 will be odd.

From the above analysis, we can see that for the product to have a last digit of 1, 3, 5, or 7, at least one of the four selected numbers must have a last digit of 1, 3, 5, or 7.

Let's calculate the probability for each case:

- Probability of selecting a number with a last digit of 1, 3, 5, or 7: 4/10 (there are four numbers with these last digits out of a total of 10 possible digits).

- Probability of selecting a number with a last digit other than 1, 3, 5, or 7: 6/10.

Since we need at least one number with a last digit of 1, 3, 5, or 7, we can use the complement rule to calculate the probability that none of the four selected numbers have a last digit of 1, 3, 5, or 7:

- Probability of none of the four numbers having a last digit of 1, 3, 5, or 7: (6/10)^4

Therefore, the probability that at least one of the four selected numbers has a last digit of 1, 3, 5, or 7 is:

1 - (6/10)^4 = 1 - 1296/10000 = 8704/10000 = 2176/2500.

To simplify the fraction, we can divide both the numerator and denominator by 16:

2176/2500 = (16 * 136)/(16 * 156.25) = 136/156.25.

Finally, we can convert this fraction to a decimal and round it to the nearest hundredth:

136/156.25 ≈ 0.8704 ≈ 0.87.

Therefore, the probability that the last digit of the product of four selected numbers is 1, 3, 5, or 7 is approximately 0.87, which is equivalent to 16/625.