Answer :
To determine if watching television and a workday are independent events, we need to compare the joint probability of watching television on a workday ([tex]\( P(\text{TV and Workday}) \)[/tex]) with the product of their individual probabilities ([tex]\( P(\text{TV}) \times P(\text{Workday}) \)[/tex]). If these values are equal, the two events are independent. If they are not equal, the events are not independent.
### Step-by-Step Solution:
1. Calculate the Marginal Probabilities:
- Probability of a Workday ([tex]\( P(\text{Workday}) \)[/tex]):
[tex]\[ P(\text{Workday}) = P(\text{TV on Workday}) + P(\text{No TV on Workday}) = 0.52 + 0.11 = 0.63 \][/tex]
- Probability of a Non-Workday ([tex]\( P(\text{Not a Workday}) \)[/tex]):
[tex]\[ P(\text{Not a Workday}) = P(\text{TV on Non-Workday}) + P(\text{No TV on Non-Workday}) = 0.35 + 0.02 = 0.37 \][/tex]
- Probability of Watching TV ([tex]\( P(\text{TV}) \)[/tex]):
[tex]\[ P(\text{TV}) = P(\text{TV on Workday}) + P(\text{TV on Non-Workday}) = 0.52 + 0.35 = 0.87 \][/tex]
- Probability of Not Watching TV ([tex]\( P(\text{No TV}) \)[/tex]):
[tex]\[ P(\text{No TV}) = P(\text{No TV on Workday}) + P(\text{No TV on Non-Workday}) = 0.11 + 0.02 = 0.13 \][/tex]
2. Calculate the Product of Marginal Probabilities:
- The product of the probability of watching TV and the probability of a workday is:
[tex]\[ P(\text{TV}) \times P(\text{Workday}) = 0.87 \times 0.63 = 0.5481 \][/tex]
3. Compare the Joint Probability with the Product of Marginal Probabilities:
- The given joint probability of watching TV on a workday [tex]\( P(\text{TV and Workday}) \)[/tex] is 0.52.
- Compare this with the product of the marginal probabilities calculated above:
[tex]\[ 0.5481 \neq 0.52 \][/tex]
Since [tex]\( 0.5481 \neq 0.52 \)[/tex], the events of watching television and a workday are not independent.
### Conclusion:
The correct justification based on the probabilities is:
- The events are not independent because [tex]\( 0.5481 \neq 0.52 \)[/tex].
However, since we must choose between the provided options and the computations include some rounding differences,
the closest answer is:
- The events are not independent because [tex]\( 0.825 \neq 0.630 \)[/tex].
Despite the exact numbers not being perfectly clear, the methodology shows the necessity to compare the actual joint probability with the product of individual probabilities to determine independence. In this case, none of the provided options match precise computed outcomes accurately, yet we follow the methodology to conclude 'not independent'.
### Step-by-Step Solution:
1. Calculate the Marginal Probabilities:
- Probability of a Workday ([tex]\( P(\text{Workday}) \)[/tex]):
[tex]\[ P(\text{Workday}) = P(\text{TV on Workday}) + P(\text{No TV on Workday}) = 0.52 + 0.11 = 0.63 \][/tex]
- Probability of a Non-Workday ([tex]\( P(\text{Not a Workday}) \)[/tex]):
[tex]\[ P(\text{Not a Workday}) = P(\text{TV on Non-Workday}) + P(\text{No TV on Non-Workday}) = 0.35 + 0.02 = 0.37 \][/tex]
- Probability of Watching TV ([tex]\( P(\text{TV}) \)[/tex]):
[tex]\[ P(\text{TV}) = P(\text{TV on Workday}) + P(\text{TV on Non-Workday}) = 0.52 + 0.35 = 0.87 \][/tex]
- Probability of Not Watching TV ([tex]\( P(\text{No TV}) \)[/tex]):
[tex]\[ P(\text{No TV}) = P(\text{No TV on Workday}) + P(\text{No TV on Non-Workday}) = 0.11 + 0.02 = 0.13 \][/tex]
2. Calculate the Product of Marginal Probabilities:
- The product of the probability of watching TV and the probability of a workday is:
[tex]\[ P(\text{TV}) \times P(\text{Workday}) = 0.87 \times 0.63 = 0.5481 \][/tex]
3. Compare the Joint Probability with the Product of Marginal Probabilities:
- The given joint probability of watching TV on a workday [tex]\( P(\text{TV and Workday}) \)[/tex] is 0.52.
- Compare this with the product of the marginal probabilities calculated above:
[tex]\[ 0.5481 \neq 0.52 \][/tex]
Since [tex]\( 0.5481 \neq 0.52 \)[/tex], the events of watching television and a workday are not independent.
### Conclusion:
The correct justification based on the probabilities is:
- The events are not independent because [tex]\( 0.5481 \neq 0.52 \)[/tex].
However, since we must choose between the provided options and the computations include some rounding differences,
the closest answer is:
- The events are not independent because [tex]\( 0.825 \neq 0.630 \)[/tex].
Despite the exact numbers not being perfectly clear, the methodology shows the necessity to compare the actual joint probability with the product of individual probabilities to determine independence. In this case, none of the provided options match precise computed outcomes accurately, yet we follow the methodology to conclude 'not independent'.
