If $P(A \cup B)=0.85, P(B)=0.50$ and $P(A \cap B)=0.30$. Then $P\left(A \cap B^{\prime}\right)=$

A company is migrating its database, and two software engineers, Ishaan and Kavya, take turns running a data-sync script that has a constant success rate of $\frac{3}{8}$ per attempt.

If Ishaan initiates the first attempt and they persist until the migration is successful, what is the probability that Kavya is the one who initiates the successful sync?

Samhita faces a three-headed dragon. She wins a "Tactical medal" if she manages to defeat exactly one of the three heads.

The battle proceeds head-by-head under the following conditions:

• The probability of defeating the first head is $\frac{\mathbf{1}}{\mathbf{3}}$.

• After a win: if she defeats a head, the probability of defeating the next head is $\frac{2}{3}$.

• After a loss: if she fails to defeat a head, the probability of defeating the next head is $\frac{\mathbf{1}}{\mathbf{4}}$.

What is the probability that Samhita earns the "Tactical medal"?

The odds against Arjun solving a problem are $\mathbf{5 : 2}$ and the odds in favour of Bhavana solving the same problem are 3:4. What is the probability that the problem is NOT solved by either of them?