Question 40 of 55intermediate🔍 AnalyzeShort Answer2 marks

Events E and F are such that P(not E or not F) = 0.25, State whether E and F are mutually exclusive.

Correct Answer

E and F are not mutually exclusive.

Given: P(not E or not F) = 0.25

We know that P(not E or not F) = P(E' ∪ F') = P((E ∩ F)')

From the context, for any event, P(E') = 1 - P(E).

Therefore, P((E ∩ F)') = 1 - P(E ∩ F)

So, 1 - P(E ∩ F) = 0.25

This gives: P(E ∩ F) = 0.75

Since P(E ∩ F) = 0.75 ≠\neq 0, the events E and F have a non-zero probability of occurring together. Hence, E and F are not mutually exclusive.

Exercise: EXERCISE 14.2 | Q: 16 | (Page Page 17)
For More Understanding

Explanation

This question tests understanding of complementary events and mutually exclusive events. From the context, we know that P(E') = 1 - P(E) for complementary events. Using De Morgan's law, 'not E or not F' equals the complement of 'E and F'. Since P(not E or not F) = 0.25, we get P(E ∩ F) = 0.75, which is non-zero. Mutually exclusive events must have P(E ∩ F) = 0, so E and F cannot be mutually exclusive.

Solution Steps

  1. Step 1: Identify that P(not E or not F) = P(E' ∪ F') = P((E ∩ F)')

  2. Step 2: Apply the complement rule from context: P((E ∩ F)') = 1 - P(E ∩ F)

  3. Step 3: Set up equation: 1 - P(E ∩ F) = 0.25

  4. Step 4: Solve: P(E ∩ F) = 0.75

  5. Step 5: Since P(E ∩ F) ≠\neq 0, events are NOT mutually exclusive

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