To solve this, let's analyze the given probabilities and express them in terms of probabilities of $A$ and $B$ and their complements.

Step 1: Analyze Given Probabilities

1. Given: $P(A \cup B') = 0.23$

   • Using the formula for the probability of a union: $P(A \cup B') = P(A) + P(B') - P(A \cap B')$

2. Given: $P(A' \cup B') = 0.81$

   • Similarly, we apply the formula for a union here: $P(A' \cup B') = P(A') + P(B') - P(A' \cap B')$
   • Since $P(A') = 1 - P(A)$ and $P(B') = 1 - P(B)$, we can rewrite this as: $P(A' \cup B') = (1 - P(A)) + (1 - P(B)) - P(A' \cap B')$
   • Simplifying further: $0.81 = 1 - P(A) - P(B) + P(A \cap B)$

Step 2: Express $P(B)$ in Terms of $P(A)$

From the two given equations, we can isolate and solve for $P(B)$ by solving a system of equations using ( P(A) \