1. Understand the experiment

Each draw is with replacement, so the probability of drawing an $A$ or $B$ remains constant at every draw:

$P(A)=\frac{10}{20}=\frac12, \qquad P(B)=\frac{10}{20}=\frac12.$

We continue drawing until either:

• the second $A$ appears, or
• the third $B$ appears.

We need the probability that the second $A$ appears before the third $B$.

2. Reformulate the condition

The event “second $A$ appears before third $B$” means that before we get 3 $B$'s, we must have already obtained 2 $A$'s.

This will certainly be decided within the first 4 draws, because:

• to get 2 $A$'s, at least 2 draws are needed,
• to get 3 $B$'s, at least 3 draws are needed.

A clean way is to look at the first 4 draws.

If the second $A$ appears before the third $B$, then among the first 4 draws there must be at least 2 A's.

Equivalently, we can count sequences where the second $A$ occurs before the third $B$.

3. Case-wise counting

Since each draw is independently $A$ or $B$ with probability $\frac12$, every length-$n$ sequence has probability $\left(\frac12\right)^n$.

We stop as soon as either 2 $A$'s or 3 $B$'s occur.

Case 1: Second $A$ occurs on draw 2

This happens only for:

$AA$

Probability:

$P(AA)=\left(\frac12\right)^2=\frac14.$

Case 2: Second $A$ occurs on draw 3

Then in the first 2 draws, there must be exactly one $A$ and one $B$, and the 3rd draw must be $A$:

$ABA,\; BAA$

Probability:

$2\cdot \left(\frac12\right)^3=\frac{2}{8}=\frac14.$

Case 3: Second $A$ occurs on draw 4

Then in the first 3 draws, there must be exactly one $A$ and two $B$'s, and the 4th draw must be $A$.

Possible sequences:

$ABBA,\; BABA,\; BBAA$

Probability:

$3\cdot \left(\frac12\right)^4=\frac{3}{16}.$

4. Add the favorable probabilities

$P(\text{2nd } A \text{ before 3rd } B)=\frac14+\frac14+\frac{3}{16}.$

Convert to denominator 16:

$\frac14=\frac{4}{16}.$

So,

$P=\frac{4}{16}+\frac{4}{16}+\frac{3}{16}=\frac{11}{16}.$

5. Check options

The correct option is:

$\boxed{\frac{11}{16}}$

So, Option B is correct.

6. Comparison with stored answer

Stored correct answer: B

Our derived answer: B

They agree.