1. Interpret the statement

We need the probability that the block '10' is followed by '01'.

This means the 4-digit pattern is: $1001$

So we need:

• 1st digit = 1
• 2nd digit = 0
• 3rd digit = 0
• 4th digit = 1

2. Given probabilities by position parity

• At odd places, probability of getting 0 is: $P(0\text{ at odd place})=\frac13$ Hence, $P(1\text{ at odd place})=1-\frac13=\frac23$

• At even places, probability of getting 0 is: $P(0\text{ at even place})=\frac12$ Hence, $P(1\text{ at even place})=1-\frac12=\frac12$

3. Apply to positions 1, 2, 3, 4

The pattern is $1001$.

• Position 1 is odd, so $P(1\text{ at 1st place})=\frac23$
• Position 2 is even, so $P(0\text{ at 2nd place})=\frac12$
• Position 3 is odd, so $P(0\text{ at 3rd place})=\frac13$
• Position 4 is even, so $P(1\text{ at 4th place})=\frac12$

Assuming independent generation at different places, required probability is: $\frac23\cdot\frac12\cdot\frac13\cdot\frac12$

4. Calculate

$\frac23\cdot\frac12=\frac13$

So, $\frac13\cdot\frac13\cdot\frac12=\frac19\cdot\frac12=\frac1{18}$

Thus, the required probability is: $\boxed{\frac1{18}}$

5. Check options

• A: $\frac1{18}$ ✅
• B: $\frac13$
• C: $\frac19$
• D: $\frac16$

So the correct option is A.