Let's analyze the information given to determine the probabilities.

Let:

• $T$ be the event of transformer damage.
• $L$ be the event of line damage.

The probabilities are provided as follows:

• $P(T) = 0.05$ (probability of transformer damage),
• $P(L) = 0.80$ (probability of line damage),
• $P(T \cap L) = 0.01$ (probability of both transformer and line damage).

(i) Probability of Transformer Damage or Line Damage

To find the probability that a given power failure involves either transformer damage or line damage (event $T \cup L$), we use the formula for the union of two events:

$P(T \cup L) = P(T) + P(L) - P(T \cap L)$

Substitute the values:

$P(T \cup L) = 0.05 + 0.80 - 0.01 = 0.84$

So, the probability that a given power failure involves either transformer damage or line damage is 0.84 or 84%.

(ii) Probability of Transformer Damage but Not Line Damage

To find the probability that a power failure involves transformer damage but not line damage (event $T$ and not $L$), we calculate $P(T \cap L^c)$, where $L^c$ is the complement of $L$.

The probability of $T$ and not $L$ can be found by subtracting the probability of both transformer and line damage from the probability of transformer damage:

$P(T \cap L^c) = P(T) - P(T \cap L)$

Substitute the values:

$P(T \cap L^c) = 0.05 - 0.01 = 0.04$

Thus, the probability that a power failure involves transformer damage but not line damage is 0.04 or 4%.

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Would you like further explanation or examples on any of these calculations?

Related Questions

1. What is the probability that a power failure involves neither transformer nor line damage?
2. How would the probabilities change if both problems occurred in 5% of cases?
3. What is the probability of only line damage occurring?
4. If 1000 power failures occurred, how many would likely involve both transformer and line damage?
5. How would you calculate the probability of a power failure involving any other type of problem?

Tip: When dealing with probabilities of combined events, always check if events are mutually exclusive or not.