Question

A person has taken the contract of a construction work. The probability of strike is 0.65. The probabilities of the construction work being completed on time in the circumstances of no strike and strike are respectively 0.80 and 0.32. Find the probability of the construction work being completed on time.

Hint:

Problems involving conditional events and partitions of the sample space (like 'strike' and 'no strike') are often solved using the Law of Total Probability or Bayes' Theorem. Identify the events and conditional probabilities clearly before applying the formula.

Answer 1

Step 1: Understanding the Concept:
This problem can be solved using the Law of Total Probability. The event of the work being completed on time depends on whether there is a strike or not. We need to consider both scenarios (strike and no strike), calculate the probability of completion in each case, and then sum them up to get the overall probability of completion.
Step 2: Key Formula or Approach:
Let S be the event that a strike occurs.
Let S' be the event that there is no strike.
Let C be the event that the construction work is completed on time.
The Law of Total Probability states: \[ P(C) = P(S) \cdot P(C|S) + P(S') \cdot P(C|S') \] where \(P(C|S)\) is the probability of completion given a strike, and \(P(C|S')\) is the probability of completion given no strike.
Step 3: Detailed Explanation:
From the problem statement, we have the following probabilities:
Probability of a strike, \(P(S) = 0.65\).
Probability of no strike, \(P(S') = 1 - P(S) = 1 - 0.65 = 0.35\).
Probability of completion given no strike, \(P(C|S') = 0.80\).
Probability of completion given a strike, \(P(C|S) = 0.32\).
Now, we apply the Law of Total Probability: \[ P(C) = P(S) \cdot P(C|S) + P(S') \cdot P(C|S') \] Substitute the given values into the formula: \[ P(C) = (0.65 \times 0.32) + (0.35 \times 0.80) \] \[ P(C) = 0.208 + 0.280 \] \[ P(C) = 0.488 \] Step 4: Final Answer:
The probability of the construction work being completed on time is 0.488.