Final answer:
The probability of the union of events E and F depends on whether they are mutually exclusive. If mutually exclusive, the union's probability is the sum of their individual probabilities. If not, the intersection must be subtracted from the sum to find the union's probability.
Step-by-step explanation:
Probability of Combined Events
To find the probability of the union of two events E and F, we can use the formula: P(E ∪ F) = P(E) + P(F) - P(E ∩ F). This is known as the Principle of Inclusion-Exclusion. If E and F are mutually exclusive events, it means that they cannot occur simultaneously, so P(E ∩ F) = 0. Therefore, for mutually exclusive events, the probability of their union is simply the sum of their individual probabilities: P(E ∪ F) = P(E) + P(F). Given that P(E) = 0.2 and P(F) = 0.65, and they are mutually exclusive, P(E ∪ F) = 0.2 + 0.65 = 0.85. However, when they are not mutually exclusive, the intersection probability must be subtracted to avoid double-counting the cases where both events occur.
For the first question, where P(E ∩ F) = 0.13, we calculate:
P(E ∪ F) = 0.2 + 0.65 - 0.13 = 0.72.
For the second question, given that P(E ∪ F) = 0.47, and we already have P(E) and P(F) values, we use them to find P(E ∩ F):
P(E ∩ F) = P(E) + P(F) - P(E ∪ F) = 0.2 + 0.65 - 0.47 = 0.38.
If E and F are mutually exclusive, as stated in the final questions, then P(E ∩ F) = 0 and P(E ∪ F) = P(E) + P(F). Thus, the probability of their intersection is zero, and the probability of their union is the sum of their individual probabilities, P(E ∪ F) = 0.2 + 0.65 = 0.85.