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

At an automated agricultural sorting facility, harvested apples are inspected for two independent quality conditions: surface bruising and internal core rot. The probability that a randomly chosen apple has surface bruising is 16\frac{1}{6}, and the probability that it has internal core rot is 15\frac{1}{5}. What is the probability that a randomly chosen apple has at least one of these two quality conditions?

  1. A
    130\frac{1}{30}
  2. B
    211\frac{2}{11}
  3. 13\frac{1}{3}Answer
  4. D
    1130\frac{11}{30}
  5. E
    25\frac{2}{5}

Answer

13\frac{1}{3}
The probability of at least one of two independent events occurring is given by P(B∪R)=P(B)+P(R)−P(B∩R)P(B \cup R) = P(B) + P(R) - P(B \cap R). Since the events are independent, P(B∩R)=P(B)×P(R)=16×15=130P(B \cap R) = P(B) \times P(R) = \frac{1}{6} \times \frac{1}{5} = \frac{1}{30}. Substituting these values gives 16+15−130=530+630−130=1030=13\frac{1}{6} + \frac{1}{5} - \frac{1}{30} = \frac{5}{30} + \frac{6}{30} - \frac{1}{30} = \frac{10}{30} = \frac{1}{3}. Alternatively, using the complement rule, P(at least one)=1−P(neither)=1−(1−16)(1−15)=1−(56×45)=1−46=13P(\text{at least one}) = 1 - P(\text{neither}) = 1 - \left(1 - \frac{1}{6}\right)\left(1 - \frac{1}{5}\right) = 1 - \left(\frac{5}{6} \times \frac{4}{5}\right) = 1 - \frac{4}{6} = \frac{1}{3}.

Step-by-Step Solution

1
Identify the given probabilities and event relationship.
Let BB be the event that an apple has surface bruising, so P(B)=16P(B) = \frac{1}{6}. Let RR be the event that an apple has internal core rot, so P(R)=15P(R) = \frac{1}{5}. The events are given as independent.
Establishing the mathematical definitions and given conditions is necessary before applying probability rules.
2
Calculate the joint probability of both events occurring, P(B∩R)P(B \cap R).
P(B∩R)=P(B)×P(R)=16×15=130P(B \cap R) = P(B) \times P(R) = \frac{1}{6} \times \frac{1}{5} = \frac{1}{30}.
For independent events, the probability that both occur simultaneously is the product of their individual probabilities.
3
Apply the addition rule of probability (inclusion-exclusion principle) to find P(B∪R)P(B \cup R).
P(B∪R)=P(B)+P(R)−P(B∩R)=16+15−130=530+630−130=1030=13P(B \cup R) = P(B) + P(R) - P(B \cap R) = \frac{1}{6} + \frac{1}{5} - \frac{1}{30} = \frac{5}{30} + \frac{6}{30} - \frac{1}{30} = \frac{10}{30} = \frac{1}{3}.
The probability of at least one event occurring requires subtracting the intersection so that the overlapping outcome is not counted twice.

Key Concept

Probability of the Union of Independent Events
Estimated Time:1m 30s
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Topics

Probability of Independent, Dependent, and Mutually Exclusive Events

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