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To determine which of the given tables could represent a conditional relative frequency table, let's analyze the properties typically associated with such tables. A conditional relative frequency table shows the probability of a certain event, given the condition that another event has occurred. This means that the sum of the probabilities across the given condition must equal 1.
Let's examine each table to identify which fits this description:
### Table 1
[tex]\[ \begin{tabular}{|c|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & A & B & Total \\ \hline C & 0.25 & 0.25 & 0.50 \\ \hline D & 0.25 & 0.25 & 0.50 \\ \hline Total & 0.50 & 0.50 & 1.0 \\ \hline \end{tabular} \][/tex]
Let's check the subtotal values:
- For event C: [tex]\( 0.25 + 0.25 = 0.50 \)[/tex]
- For event D: [tex]\( 0.25 + 0.25 = 0.50 \)[/tex]
- Totals for A and B: [tex]\( 0.50 + 0.50 = 1.0 \)[/tex]
In this table, the sums across both the rows and columns match, verifying that the total probabilities add up to 1.0. Therefore, this table could be a conditional relative frequency table.
### Table 2
[tex]\[ \begin{tabular}{|c|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & A & B & Total \\ \hline C & 0.25 & 0.75 & 1.0 \\ \hline D & 0.35 & 0.65 & 1.0 \\ \hline Total & 0.30 & 0.70 & 1.0 \\ \hline \end{tabular} \][/tex]
Let's check the subtotal values:
- For event C: [tex]\( 0.25 + 0.75 = 1.0 \)[/tex]
- For event D: [tex]\( 0.35 + 0.65 = 1.0 \)[/tex]
- Totals for A and B: [tex]\( 0.30 + 0.70 = 1.0 \)[/tex]
Here, the totals are as expected as well, summing to 1.0. Additionally, the rows also sum to 1.0. Therefore, this table could also be a conditional relative frequency table.
### Table 3
[tex]\[ \begin{tabular}{|c|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & A & B & Total \\ \hline C & 0.75 & 0.25 & 0.50 \\ \hline \end{tabular} \][/tex]
Here, we only have one row to consider:
- For event C: [tex]\( 0.75 + 0.25 = 1.0 \)[/tex]
Even though the total is 1.0 for a single event (C), it does not show the totals down a column or across rows for a second event.
### Conclusion:
Both the first and second tables could be conditional relative frequency tables. However, based on completeness and easier visibility, Table 2 exhibits a clearer representation of conditional probabilities summed to 1:
[tex]\[ \begin{tabular}{|c|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & A & B & Total \\ \hline C & 0.25 & 0.75 & 1.0 \\ \hline D & 0.35 & 0.65 & 1.0 \\ \hline Total & 0.30 & 0.70 & 1.0 \\ \hline \end{tabular} \][/tex]
Let's examine each table to identify which fits this description:
### Table 1
[tex]\[ \begin{tabular}{|c|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & A & B & Total \\ \hline C & 0.25 & 0.25 & 0.50 \\ \hline D & 0.25 & 0.25 & 0.50 \\ \hline Total & 0.50 & 0.50 & 1.0 \\ \hline \end{tabular} \][/tex]
Let's check the subtotal values:
- For event C: [tex]\( 0.25 + 0.25 = 0.50 \)[/tex]
- For event D: [tex]\( 0.25 + 0.25 = 0.50 \)[/tex]
- Totals for A and B: [tex]\( 0.50 + 0.50 = 1.0 \)[/tex]
In this table, the sums across both the rows and columns match, verifying that the total probabilities add up to 1.0. Therefore, this table could be a conditional relative frequency table.
### Table 2
[tex]\[ \begin{tabular}{|c|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & A & B & Total \\ \hline C & 0.25 & 0.75 & 1.0 \\ \hline D & 0.35 & 0.65 & 1.0 \\ \hline Total & 0.30 & 0.70 & 1.0 \\ \hline \end{tabular} \][/tex]
Let's check the subtotal values:
- For event C: [tex]\( 0.25 + 0.75 = 1.0 \)[/tex]
- For event D: [tex]\( 0.35 + 0.65 = 1.0 \)[/tex]
- Totals for A and B: [tex]\( 0.30 + 0.70 = 1.0 \)[/tex]
Here, the totals are as expected as well, summing to 1.0. Additionally, the rows also sum to 1.0. Therefore, this table could also be a conditional relative frequency table.
### Table 3
[tex]\[ \begin{tabular}{|c|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & A & B & Total \\ \hline C & 0.75 & 0.25 & 0.50 \\ \hline \end{tabular} \][/tex]
Here, we only have one row to consider:
- For event C: [tex]\( 0.75 + 0.25 = 1.0 \)[/tex]
Even though the total is 1.0 for a single event (C), it does not show the totals down a column or across rows for a second event.
### Conclusion:
Both the first and second tables could be conditional relative frequency tables. However, based on completeness and easier visibility, Table 2 exhibits a clearer representation of conditional probabilities summed to 1:
[tex]\[ \begin{tabular}{|c|c|c|c|} \cline { 2 - 4 } \multicolumn{1}{c|}{} & A & B & Total \\ \hline C & 0.25 & 0.75 & 1.0 \\ \hline D & 0.35 & 0.65 & 1.0 \\ \hline Total & 0.30 & 0.70 & 1.0 \\ \hline \end{tabular} \][/tex]
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