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

A medical laboratory uses an automated analyzer to screen blood samples for two distinct markers, Marker A and Marker B. The probability that a randomly selected sample contains Marker A is 0.400.40, and the probability that it contains Marker B is 0.250.25. If the presence of Marker A and the presence of Marker B are independent events, what is the probability that a randomly selected sample contains at least one of these two markers?

  1. A
    0.10
  2. B
    0.45
  3. 0.55Answer
  4. D
    0.65
  5. E
    0.70

Answer

The probability that a randomly selected sample contains at least one of the two markers is 0.550.55.
To find the probability that a sample contains at least one marker, apply the general addition rule P(A or B)=P(A)+P(B)−P(A and B)P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B}). Because the events are independent, P(A and B)=P(A)×P(B)=0.40×0.25=0.10P(\text{A and B}) = P(\text{A}) \times P(\text{B}) = 0.40 \times 0.25 = 0.10. Substituting the values gives 0.40+0.25−0.10=0.550.40 + 0.25 - 0.10 = 0.55. Alternatively, using the complementary probability rule yields 1−P(neither)=1−(1−0.40)(1−0.25)=1−(0.60×0.75)=1−0.45=0.551 - P(\text{neither}) = 1 - (1 - 0.40)(1 - 0.25) = 1 - (0.60 \times 0.75) = 1 - 0.45 = 0.55.

Step-by-Step Solution

1
Identify the given probabilities and event relationship.
P(A)=0.40P(\text{A}) = 0.40, P(B)=0.25P(\text{B}) = 0.25, and events A and B are independent.
Establishing the parameters is necessary to apply the appropriate probability formulas.
2
Calculate the joint probability P(A and B)P(\text{A and B}).
P(A and B)=P(A)×P(B)=0.40×0.25=0.10P(\text{A and B}) = P(\text{A}) \times P(\text{B}) = 0.40 \times 0.25 = 0.10.
For independent events, the probability of both events occurring simultaneously is the product of their individual probabilities.
3
Apply the general addition rule for probability to find P(A or B)P(\text{A or B}).
P(A or B)=P(A)+P(B)−P(A and B)=0.40+0.25−0.10=0.55P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B}) = 0.40 + 0.25 - 0.10 = 0.55.
The probability of at least one event occurring requires subtracting the overlapping joint probability to avoid double-counting.

Key Concept

Probability of Independent Events and the General Addition Rule
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Topics

Probability of Independent, Dependent, and Mutually Exclusive Events

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