Based on the image provided, we are working with a tree diagram and probabilities table for events $A$ and $B$. Let’s break this down step by step:

1. Extract Information from the Table:

The table below the diagram shows counts for combinations of $A, A', B, B'$:

• $A \cap B = 21$,
• $A \cap B' = 18$,
• $A' \cap B = 24$,
• $A' \cap B' = 57$.

2. Calculate Total Outcomes:

Sum all values: $\text{Total} = 21 + 18 + 24 + 57 = 120.$

3. Fill in the Probabilities:

• $P(A)$: Probability of event $A$: $P(A) = \frac{\text{Number of outcomes in } A}{\text{Total outcomes}} = \frac{21 + 18}{120} = \frac{39}{120} = 0.325.$

• $P(A')$: Complement of $P(A)$: $P(A') = 1 - P(A) = 1 - 0.325 = 0.675.$

• $P(B)$: Probability of event $B$: $P(B) = \frac{\text{Number of outcomes in } B}{\text{Total outcomes}} = \frac{21 + 24}{120} = \frac{45}{120} = 0.375.$

• $P(B')$: Complement of $P(B)$: $P(B') = 1 - P(B) = 1 - 0.375 = 0.625.$

• $P(B | A)$: Conditional probability of $B$ given $A$: $P(B | A) = \frac{P(A \cap B)}{P(A)} = \frac{\frac{21}{120}}{0.325} = \frac{21}{39} = 0.538.$

• $P(B | A')$: Conditional probability of $B$ given $A'$: $P(B | A') = \frac{P(A' \cap B)}{P(A')} = \frac{\frac{24}{120}}{0.675} = \frac{24}{81} = 0.296.$

• $P(B') | A$: Probability of $B'$ given $A$: $P(B' | A) = 1 - P(B | A) = 1 - 0.538 = 0.462.$

• $P(B') | A'$: Probability of $B'$ given $A'$: $P(B' | A') = 1 - P(B | A') = 1 - 0.296 = 0.704.$

4. Test for Independence:

For $A$ and $B$ to be independent, $P(B | A) = P(B)$. Since $P(B | A) = 0.538$ and $P(B) = 0.375$, they are not independent because these values are not equal.

Would you like me to organize this into a detailed table for clarity? Let me know!