To determine which contractor Anne should choose to maximize the probability that the job will be done on time and on budget, we need to consider the combined probability of both conditions being met for each contractor. This involves calculating the joint probability for each contractor, given that these events are independent.
Let's break it down step by step:
1. Identify the given probabilities:
- For Contractor A:
- Probability that the job is done on time: [tex]\( P_{A,\text{time}} = 0.97 \)[/tex]
- Probability that the job is done within budget: [tex]\( P_{A,\text{budget}} = 0.96 \)[/tex]
- For Contractor B:
- Probability that the job is done on time: [tex]\( P_{B,\text{time}} = 0.93 \)[/tex]
- Probability that the job is done within budget: [tex]\( P_{B,\text{budget}} = 0.98 \)[/tex]
2. Calculate the combined probability for Contractor A:
Since the probabilities are independent, the combined probability for Contractor A can be found by multiplying the individual probabilities:
[tex]\[
P_{A,\text{combined}} = P_{A,\text{time}} \times P_{A,\text{budget}}
\][/tex]
Substituting the given probabilities:
[tex]\[
P_{A,\text{combined}} = 0.97 \times 0.96 = 0.9312
\][/tex]
3. Calculate the combined probability for Contractor B:
Similarly, for Contractor B, the combined probability is:
[tex]\[
P_{B,\text{combined}} = P_{B,\text{time}} \times P_{B,\text{budget}}
\][/tex]
Substituting the given probabilities:
[tex]\[
P_{B,\text{combined}} = 0.93 \times 0.98 = 0.9114
\][/tex]
4. Compare the combined probabilities:
- Contractor A: [tex]\( 0.9312 \)[/tex]
- Contractor B: [tex]\( 0.9114 \)[/tex]
Since 0.9312 is greater than 0.9114, Contractor A has a higher combined probability of completing the job on time and within budget.
Therefore, Anne should choose Contractor A to maximize the probability that the job will be done on time and on budget.
