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Zorluk: KolayProbability of Independent, Dependent, and Mutually Exclusive Events

Events EE and FF are two events in a sample space such that P(E)=0.30P(E) = 0.30 and P(F)=0.40P(F) = 0.40. Which of the following statements must be true? Select all that apply.

  1. If EE and FF are mutually exclusive, then P(E and F)=0P(E \text{ and } F) = 0.Cevap
  2. If EE and FF are independent, then P(E and F)=0.12P(E \text{ and } F) = 0.12.Cevap
  3. C
    If EE and FF are independent, then P(E or F)=0.70P(E \text{ or } F) = 0.70.
  4. D
    If EE and FF are mutually exclusive, then EE and FF are independent.
  5. E
    If EE and FF are independent, then P(E given F)=0.12P(E \text{ given } F) = 0.12.

Cevap

The correct statements are that if EE and FF are mutually exclusive, then P(E and F)=0P(E \text{ and } F) = 0, and if EE and FF are independent, then P(E and F)=0.12P(E \text{ and } F) = 0.12.
The statement asserting that P(E and F)=0P(E \text{ and } F) = 0 for mutually exclusive events is correct because mutually exclusive events by definition cannot occur together. The statement asserting that P(E and F)=0.12P(E \text{ and } F) = 0.12 for independent events is correct because independent events satisfy the multiplication rule P(E and F)=P(E)×P(F)=0.30×0.40=0.12P(E \text{ and } F) = P(E) \times P(F) = 0.30 \times 0.40 = 0.12.

Adım Adım Çözüm

1
Evaluate the statement regarding mutually exclusive events.
P(E and F)=0P(E \text{ and } F) = 0.
By definition, mutually exclusive events cannot both happen at the same time, so P(E and F)=0P(E \text{ and } F) = 0 is true.
2
Evaluate the joint probability for independent events.
P(E and F)=P(E)×P(F)=0.30×0.40=0.12P(E \text{ and } F) = P(E) \times P(F) = 0.30 \times 0.40 = 0.12.
The multiplication rule for independent events states that P(E and F)=P(E)⋅P(F)P(E \text{ and } F) = P(E) \cdot P(F).
3
Evaluate the union probability P(E or F)P(E \text{ or } F) for independent events.
P(E or F)=P(E)+P(F)−P(E and F)=0.30+0.40−0.12=0.58P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F) = 0.30 + 0.40 - 0.12 = 0.58.
Simply adding P(E)+P(F)=0.70P(E) + P(F) = 0.70 fails to subtract the intersection that is counted twice.
4
Evaluate conditional probability for independent events.
P(E given F)=P(E)=0.30P(E \text{ given } F) = P(E) = 0.30.
Independence implies that knowing event FF occurred does not change the probability of event EE.

Anahtar Kavram

Probability rules for mutually exclusive and independent events
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Konular

Probability of Independent, Dependent, and Mutually Exclusive Events

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