Calculating the Probability of Construction Job Delay

The question asks for the probability that a construction job will not be completed on time. We are given probabilities related to a strike and the impact of a strike (or no strike) on the job completion time.

Let's define the events:

• \(S\): The event that there is a strike.
• \(S^c\): The event that there is no strike.
• \(C\): The event that the construction job is completed on time.
• \(C^c\): The event that the construction job is not completed on time.

We are given the following probabilities:

• Probability of strike, \(P(S) = 0.6\).
• Probability of no strike, \(P(S^c)\). Since having a strike or not having a strike are complementary events, \(P(S^c) = 1 - P(S) = 1 - 0.6 = 0.4\).
• Probability of completion on time given no strike, \(P(C | S^c) = 0.85\). This is a conditional probability.
• Probability of completion on time given strike, \(P(C | S) = 0.35\). This is also a conditional probability.

We need to find the probability that the construction job will not be completed on time, which is \(P(C^c)\).

We can find \(P(C^c)\) by first finding the probability that the construction job is completed on time, \(P(C)\), and then using the fact that \(C\) and \(C^c\) are complementary events:

\(P(C^c) = 1 - P(C)\)

Applying the Law of Total Probability

The events \(S\) (strike) and \(S^c\) (no strike) are mutually exclusive (they cannot happen at the same time) and exhaustive (one of them must happen). In such cases, we can use the Law of Total Probability to find the probability of an event \(C\) that depends on these events:

\[P(C) = P(C | S) \cdot P(S) + P(C | S^c) \cdot P(S^c)\]

Let's plug in the given values:

• \(P(S) = 0.6\)
• \(P(S^c) = 0.4\)
• \(P(C | S) = 0.35\)
• \(P(C | S^c) = 0.85\)

Now, calculate \(P(C)\):

\[P(C) = (0.35 \times 0.6) + (0.85 \times 0.4)\] \[P(C) = 0.21 + 0.34\] \[P(C) = 0.55\]

So, the probability that the construction job is completed on time is 0.55.

Calculating the Probability of Not Completing on Time

We want to find \(P(C^c)\), the probability that the construction job will not be completed on time. Since \(C\) and \(C^c\) are complementary events, their probabilities sum to 1:

\[P(C^c) = 1 - P(C)\]

Using the value of \(P(C) = 0.55\):

\[P(C^c) = 1 - 0.55\] \[P(C^c) = 0.45\]

Therefore, the probability that the construction job will not be completed on time is 0.45.

Revision Table: Construction Job Probability

Additional Information: Probability Concepts

This problem utilizes several fundamental concepts in probability:

• Conditional Probability: The probability of an event occurring given that another event has already occurred. \(P(A|B)\) is the probability of event A happening given event B has happened.
• Mutually Exclusive Events: Events that cannot occur at the same time. In this problem, 'strike' and 'no strike' are mutually exclusive.
• Exhaustive Events: A set of events is exhaustive if at least one of the events must occur. 'Strike' and 'no strike' are exhaustive because either there is a strike or there isn't.
• Law of Total Probability: This rule is used to find the total probability of an event by considering all possible ways it can occur through different, mutually exclusive scenarios. If \(B_1, B_2, \dots, B_n\) are mutually exclusive and exhaustive events, then for any event A: \(P(A) = \sum_{i=1}^{n} P(A | B_i) P(B_i)\).
• Complementary Events: Two events are complementary if they are mutually exclusive and exhaustive. If A is an event, its complement \(A^c\) is the event that A does not occur. \(P(A^c) = 1 - P(A)\).

Understanding these concepts is crucial for solving problems involving multiple events and dependencies, like the construction job delay example.