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Difficulty: MediumProbability of Independent, Dependent, and Mutually Exclusive Events

A laboratory tests two solar panels, Panel A and Panel B, under identical conditions. The probability that Panel A operates at peak efficiency on any given day is 0.750.75, and the probability that Panel B operates at peak efficiency on any given day is 0.600.60. The daily efficiency outcomes of the two panels are independent events. Which of the following statements must be true? Select all such statements.

  1. The probability that both panels operate at peak efficiency on a given day is 0.450.45.Answer
  2. B
    The probability that neither panel operates at peak efficiency on a given day is 0.350.35.
  3. The probability that at least one panel operates at peak efficiency on a given day is 0.900.90.Answer
  4. D
    The conditional probability that Panel A operates at peak efficiency, given that Panel B operates at peak efficiency, is 0.450.45.
  5. The probability that Panel A operates at peak efficiency and Panel B does not operate at peak efficiency on a given day is 0.300.30.Answer

Answer

The correct statements are: the probability that both panels operate at peak efficiency is 0.450.45; the probability that at least one panel operates at peak efficiency is 0.900.90; and the probability that Panel A operates at peak efficiency while Panel B does not is 0.300.30.
The statements confirming that both panels operate at peak efficiency (0.450.45), that at least one operates at peak efficiency (0.900.90), and that Panel A operates while Panel B does not (0.300.30) are mathematically sound applications of independent event rules.

Step-by-Step Solution

1
Identify given probabilities and independence condition
P(A)=0.75P(A) = 0.75, P(B)=0.60P(B) = 0.60, and events AA and BB are independent.
Establishes the given parameter values.
2
Calculate joint probability of both events occurring
P(A∩B)=P(A)×P(B)=0.75×0.60=0.45P(A \cap B) = P(A) \times P(B) = 0.75 \times 0.60 = 0.45.
For independent events, joint probability equals the product of individual probabilities.
3
Determine complement probabilities and probability of neither event occurring
P(Ac)=1−0.75=0.25P(A^c) = 1 - 0.75 = 0.25, P(Bc)=1−0.60=0.40P(B^c) = 1 - 0.60 = 0.40, so P(Ac∩Bc)=0.25×0.40=0.10P(A^c \cap B^c) = 0.25 \times 0.40 = 0.10.
Complements of independent events are also independent.
4
Calculate the union probability (at least one panel at peak efficiency)
P(A∪B)=1−P(Ac∩Bc)=1−0.10=0.90P(A \cup B) = 1 - P(A^c \cap B^c) = 1 - 0.10 = 0.90.
The event 'at least one' is the logical complement of 'neither'.
5
Evaluate conditional probability P(A∣B)P(A \mid B) and difference probability P(A∩Bc)P(A \cap B^c)
P(A∣B)=P(A)=0.75P(A \mid B) = P(A) = 0.75 and P(A∩Bc)=0.75×0.40=0.30P(A \cap B^c) = 0.75 \times 0.40 = 0.30.
Independence implies P(A∣B)=P(A)P(A \mid B) = P(A) and P(A∩Bc)=P(A)P(Bc)P(A \cap B^c) = P(A) P(B^c).

Key Concept

Probability rules for independent events, complement rule, and conditional probability definition
Estimated Time:1m 30s
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Topics

Probability of Independent, Dependent, and Mutually Exclusive Events

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