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Zorluk: OrtaProbability 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}Cevap
  4. D
    1130\frac{11}{30}
  5. E
    25\frac{2}{5}

Cevap

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}.

Adım Adım Çözüm

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.

Anahtar Kavram

Probability of the Union of Independent Events
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Konular

Probability of Independent, Dependent, and Mutually Exclusive Events

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