Final answer:
The student's question involves calculating various probabilities using conditional probability and set theory. After determining the probability of F, given E, and vice versa, and interpreting the probability of events based on intersections and unions, the calculations are made step by step. Some expressions provided in the question are either incorrect or redundant.
Step-by-step explanation:
To solve these problems, we will use the formulae of conditional probability and the principles of set theory. Given:
P(E) = 0.2, P(E ∩ F) = 0.16, and P(E ∪ F) = 0.52.
- For P(F|E), the probability of event F given that event E has occurred, use the conditional probability formula: P(F|E) = P(E ∩ F) / P(E). So, P(F|E) = 0.16 / 0.2 = 0.8.
- For P(E | F), the probability of event E given that event F has occurred, we need the value of P(F), given by P(F) = P(E ∪ F) - P(E) plus P(F). Subtracting, we get P(F) = 0.52 - 0.2 = 0.32. Then, P(E | F) = P(E ∩ F) / P(F) = 0.16 / 0.32 = 0.5.
- For P(E ∪ F) | E ∪ F), since the condition is the event itself, the probability is 1.
- For P(E | E ∪ F), it is the probability of E given E or F has occurred. Use P(E | E ∪ F) = P(E ∩ (E ∪ F)) / P(E ∪ F). Since E intersection (E or F) is just E, it simplifies to P(E). So P(E | E ∪ F) = P(E) / P(E ∪ F) = 0.2 / 0.52 = 0.3846.
- For P(E ∩ F | E ∪ F), use the formula P(E ∩ F | E ∪ F) = P(E ∩ F) / P(E ∪ F). This results in 0.16 / 0.52 = 0.3077.
- For P((E ∩ F)ϕ | E ∪ F), where ϕ represents the complement, calculate the probability of not E and F given that E or F has occurred. This can be found as 1 - P(E ∩ F | E ∪ F), which equals 1 - 0.3077 = 0.6923.
- For P((E ∪ F)P(E ∪ F)), this expression seems incorrect and not well-formed. If we assume it means finding P(E ∪ F) given itself, it's again 1.