Concept:
Conditional probability is used when one event has already occurred. If sampling is done without replacement, the total number of objects decreases after each draw.
Step 1: {Identify the initial composition of the bag:}
Initially,
\[
\text{Red balls} = 4, \qquad \text{Black balls} = 6
\]
Total balls:
\[
4+6=10
\]
Step 2: {Apply the given condition:}
The first ball drawn is black. Hence one black ball is removed.
Remaining balls:
\[
\text{Red} = 4, \qquad \text{Black} = 5
\]
Total remaining:
\[
4+5=9
\]
Step 3: {Calculate the required probability:}
Probability that the second ball is red:
\[
P(\text{Second is Red} \mid \text{First is Black})
=
\frac{4}{9}
\]