Question

Suppose that \(P(E)=0.6 .\) What can you say about \(P(E \mid F)\) when (a) \(E\) and \(F\) are mutually exclusive? (b) \(E \subset F ?\) (c) \(F \subset E ?\)

Short answer

(a) When E and F are mutually exclusive, \(P(E|F) = 0\). (b) When E is a subset of F, \(0.6 \leq P(E|F) \leq 1\). (c) When F is a subset of E, \(P(E|F) = 1\).

Step-by-step solution

(a) Mutually Exclusive Events

When two events are mutually exclusive, it means that they cannot both happen at the same time. In other words, if E happens, F cannot happen and vice versa. Mathematically, this means that the intersection of E and F is empty, denoted as E ∩ F = ∅. Therefore, P(E ∩ F) = 0. According to the definition of conditional probability, we have: P(E|F) = \(\dfrac{P(E ∩ F)}{P(F)}\) Since P(E ∩ F) = 0 in this case, the conditional probability, P(E|F), will always equal 0 because the numerator of the fraction is 0: P(E|F) = \(\dfrac{0}{P(F)}\) = 0 So, when E and F are mutually exclusive, the probability of E happening given F has happened is 0.

(b) E is a Subset of F

If event E is a subset of event F (E ⊆ F), it means that if F happens, then E must also happen. Mathematically, this means E ∩ F = E. Therefore, P(E ∩ F) = P(E). Using the definition of conditional probability, P(E|F) = \(\dfrac{P(E ∩ F)}{P(F)}\), we can substitute the values we know: P(E|F) = \(\dfrac{P(E)}{P(F)}\) Since E is a subset of F, F could be equal to or greater than E, thus P(F) ≥ P(E). In this case, P(E) = 0.6, so 0.6 ≤ P(F) ≤ 1. Therefore, the range of possible values for P(E|F) is: 1 ≥ P(E|F) ≥ \(\dfrac{0.6}{1}\) = 0.6 In summary, when event E is a subset of event F, the probability of E happening given F has happened is between 0.6 and 1, inclusive.

(c) F is a Subset of E

If event F is a subset of event E (F ⊆ E), it means that if F happens, then E must also happen. Mathematically, this means E ∩ F = F. Therefore, P(E ∩ F) = P(F). Using the definition of conditional probability, P(E|F) = \(\dfrac{P(E ∩ F)}{P(F)}\), we can plug in the values we know: P(E|F) = \(\dfrac{P(F)}{P(F)}\) Since F has some probability of happening (P(F) > 0), we can simplify the fraction to get: P(E|F) = 1 So, when event F is a subset of event E, the probability of E happening given F has happened is 1.