In a production lot of 100 LED bulbs:
- $\(\mathbf{1 5 \%}\)$ are size A , and $\(\mathbf{8 5 \%}\)$ are size B .
- 50\% are red, and 50\% are green.
- 10 bulbs are red and size $A$.

We need to find the number of bulbs that are green and size B.

Step 1: Organize the Given Data
First, let's summarize the information in a table for clarity.

\(\begin{tabular}{llll}
& Size A & Size B & Total \\
\hline Red & 10 & $?$ & 50 \\
\hline Green & $?$ & $?$ & 50 \\
\hline Total & 15 & 85 & 100 \\
\hline
\end{tabular}\)

Step 2: Fill in the Known Values
1. Total Size A bulbs:
$\(15 \%\)$ of 100 = 15 bulbs.
- Given that 10 are red and size A , the remaining green and size $\(\mathbf{A}\)$ bulbs: $\(15-10=5\)$ bulbs.
2. Total Size B bulbs:
$\(85 \%\)$ of 100 = 85 bulbs.
- We don't yet know how many are red or green.
3. Total Red bulbs:
$\(50 \%\)$ of 100 = $\(\mathbf{5 0}\)$ bulbs.
- Given that 10 are red and size $A$, the remaining red and size $B$ bulbs: $\(50-10=40\)$ bulbs.

4. Total Green bulbs:
$\(50 \%\)$ of 100 = 50 bulbs.
- We already have 5 green and size A bulbs, so the remaining green and size $\(\mathbf{B}\)$ bulbs: $\(50-5=45\)$ bulbs.

Step 3: Verify the Calculations
Let's cross-check the numbers to ensure consistency.
\(\begin{tabular}{|l|l|l|l|}
\hline & Size A & Size B & Total \\
\hline Red & 10 & 40 & 50 \\
\hline Green & 5 & 45 & 50 \\
\hline Total & 15 & 85 & 100 \\
\hline
\end{tabular}\)

- Size A Total: $\(10+5=15\)$ (matches the given $\(15 \%\)$ ).
- Size B Total: $\(40+45=85\)$ (matches the given $\(85 \%\)$ ).
- Red Total: $\(10+40=50\)$ (matches the given $\(50 \%\)$ ).
- Green Total: $\(5+45=50\)$ (matches the given $\(50 \%\)$ ).

Step 4: Answer the Question
The number of bulbs that are green and size B is $\(\mathbf{4 5}\)$.