Two events E and F in a sample space have probabilities P(E) = 0.40 and P(F) = 0.50. Which… — GRE General Test

Two events $E$ and $F$ in a sample space have probabilities $P(E) = 0.40$ and $P(F) = 0.50$ . Which of the following statements must be true? Select all such statements.

If events $E$ and $F$ are mutually exclusive, then $P(E \text{ or } F) = 0.90$ .Answer
If events $E$ and $F$ are independent, then $P(E \text{ and } F) = 0.20$ .Answer
Events $E$ and $F$ cannot be both mutually exclusive and independent.Answer
D
If events $E$ and $F$ are independent, then $P(E \text{ or } F) = 0.90$ .
E
If events $E$ and $F$ are mutually exclusive, then $P(E \text{ and } F) = 0.90$ .

Answer

The correct statements are: 'If events E and F are mutually exclusive, then P(E or F) = 0.90', 'If events E and F are independent, then P(E and F) = 0.20', and 'Events E and F cannot be both mutually exclusive and independent.'

The statement regarding mutually exclusive events correctly uses the addition rule P(E or F) = P(E) + P(F) = 0.90 because there is no overlap. The statement regarding independent events correctly applies the multiplication rule P(E and F) = P(E) * P(F) = 0.20. Finally, events with non-zero probabilities cannot be both mutually exclusive (requiring zero intersection) and independent (requiring positive intersection), making the impossibility statement correct.

Step-by-Step Solution

Evaluate the union probability for mutually exclusive events.

P(E or F) = P(E) + P(F) = 0.40 + 0.50 = 0.90.

By definition of mutually exclusive events, P(E and F) = 0.

Evaluate the joint probability for independent events.

P(E and F) = P(E) * P(F) = 0.40 * 0.50 = 0.20.

The multiplication rule applies directly to independent events.

Evaluate whether mutually exclusive events with non-zero probabilities can be independent.

They cannot be both mutually exclusive and independent.

Mutual exclusivity requires P(E and F) = 0, whereas independence requires P(E and F) = 0.20, which are contradictory.

Evaluate the union probability for independent events.

P(E or F) = 0.40 + 0.50 - 0.20 = 0.70.

The general addition rule P(E or F) = P(E) + P(F) - P(E and F) must account for the overlap.

Key Concept

Distinction between independent events and mutually exclusive events in probability

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