Question

An organisation needs to select a building construction company to make houses in cities. Three companies X, Y and Z are short-listed with the probability of their selection being \(\frac{1}{3}\), \(\frac{2}{9}\) and \(\frac{4}{9}\) respectively. It was found that there were 20%, 40% and 30% chances respectively, of delivering poor construction quality.

Based on the above information, answer the following questions :

If the building constructed was found to be strong, then what is the probability that the construction contract was given to the company X ?

Hint:

Use Bayes theorem with P(strong given X) = 4/5, P(strong given Y) = 3/5 and P(strong given Z) = 7/10.

Answer 1

Step 1: Set up the notation:
Write $X$, $Y$, $Z$ for the events that each company gets the contract. Write $G$ for a strong (good) building.
The job is to compute $P(X \mid G)$ from the given data.

Step 2: Use a table of values:
Each company has a chance of poor work, so the chance of strong work is one minus that chance.
For X it is $1-\frac{20}{100}=0.8$. For Y it is $1-0.4=0.6$. For Z it is $1-0.3=0.7$.

Step 3: Multiply selection by quality:
The joint probabilities of "company chosen and building strong" are:
\[ P(X \cap G)=\frac{1}{3}(0.8)=\frac{12}{45} \]
\[ P(Y \cap G)=\frac{2}{9}(0.6)=\frac{6}{45} \]
\[ P(Z \cap G)=\frac{4}{9}(0.7)=\frac{14}{45} \]

Step 4: Divide the share of X by the total:
The three joint probabilities add up to $\frac{32}{45}$, which is $P(G)$.
So the share of X among strong buildings is $\frac{12}{32}$.
\[ P(X \mid G)=\frac{12/45}{32/45}=\frac{3}{8} \]

Final Answer:
Given that the building is strong, the contract was with X with probability $\frac{3}{8}$. \[ \boxed{\frac{3}{8}} \]