Strands of copper wire from a manufacturer are analyzed for strength and conductivity. The results from 100 strands are as follows:
a. If a strand is randomly selected, what is the probability that its conductivity is high and its strength is high?
b. If a strand is randomly selected, what is the probability that its conductivity is low or its strength is low?
c. Consider the event that a strand has low conductivity and the event that the strand has low strength. Are these two events mutually exclusive?
The Correct Answer and Explanation is :
To address the questions regarding the copper wire strands’ strength and conductivity, we first need to interpret the provided data. The table from the source indicates the following distribution for 100 strands:
| Strength/Conductivity | High Conductivity | Low Conductivity | Total |
|---|---|---|---|
| High Strength | 74 | 11 | 85 |
| Low Strength | 8 | 7 | 15 |
| Total | 82 | 18 | 100 |
a. Probability of High Conductivity and High Strength
The probability that a randomly selected strand has both high conductivity and high strength is calculated by dividing the number of strands with both characteristics by the total number of strands:
[ P(\text{High Conductivity and High Strength}) = \frac{74}{100} = 0.74 ]
Therefore, the probability is 0.74, or 74%.
b. Probability of Low Conductivity or Low Strength
To find the probability that a strand has either low conductivity or low strength, we use the formula for the union of two events:
[ P(\text{Low Conductivity or Low Strength}) = P(\text{Low Conductivity}) + P(\text{Low Strength}) – P(\text{Low Conductivity and Low Strength}) ]
From the table:
- ( P(\text{Low Conductivity}) = \frac{18}{100} = 0.18 )
- ( P(\text{Low Strength}) = \frac{15}{100} = 0.15 )
- ( P(\text{Low Conductivity and Low Strength}) = \frac{7}{100} = 0.07 )
Substituting these values:
[ P(\text{Low Conductivity or Low Strength}) = 0.18 + 0.15 – 0.07 = 0.26 ]
Thus, the probability is 0.26, or 26%.
c. Are Low Conductivity and Low Strength Mutually Exclusive?
Two events are mutually exclusive if they cannot occur simultaneously. In this case, the number of strands with both low conductivity and low strength is 7. Since this number is greater than zero, the events are not mutually exclusive.
Explanation
- High Conductivity and High Strength: The probability of selecting a strand with both high conductivity and high strength is straightforward. Out of 100 strands, 74 exhibit both characteristics, leading to a probability of 0.74.
- Low Conductivity or Low Strength: To determine the probability of a strand having either low conductivity or low strength, we apply the principle of inclusion-exclusion. This principle ensures that we do not double-count strands that have both low conductivity and low strength. By adding the probabilities of each event and subtracting the probability of their intersection, we obtain the correct probability of 0.26.
- Mutual Exclusivity: The concept of mutual exclusivity is crucial in probability theory. Since there are strands that possess both low conductivity and low strength, the events are not mutually exclusive. This means that the occurrence of one event does not preclude the occurrence of the other.
Understanding these probabilities and the relationship between the events helps in assessing the quality and reliability of the copper wire strands. High conductivity and high strength are desirable properties for electrical conductors, as they ensure efficient current flow and durability. Conversely, low conductivity and low strength are undesirable, as they can lead to increased resistance and potential failure of the wire.
By analyzing the distribution of these properties, manufacturers can identify areas for improvement in their production processes to enhance the overall performance of their products.