Final answer:
According to the formula for conditional probability, if P(A|B) = 5/9 and P(B) = 1/3, we can find P(A|B') by finding the complement of event B and using the formula P(A|B') = (P(A AND B) + P(A AND B')) / P(B'). Plugging in the given values, we find that P(A|B') = 1.
Step-by-step explanation:
According to the formula for conditional probability, P(A|B) = P(A AND B)/P(B). Given P(A|B) = 5/9 and P(B) = 1/3, we can plug in these values to solve for P(A AND B). Rearranging the formula, we get P(A AND B) = P(A|B) * P(B) = (5/9) * (1/3) = 5/27.
Next, we can find P(B'), which represents the complement of event B. Since the sum of the probabilities of all possible outcomes in a sample space is 1, we know that P(B') = 1 - P(B) = 1 - 1/3 = 2/3.
Finally, we can use the formula for conditional probability to find P(A|B'). P(A|B') = P(A AND B') / P(B') = (P(A AND B) + P(A AND B')) / P(B') = (5/27 + P(A AND B')) / (2/3). Since these events are complements, P(A AND B') = 1 - P(A AND B) = 1 - 5/27 = 22/27. Plugging in this value, we get P(A|B') = (5/27 + 22/27) / (2/3) = 27/27 = 1.