Answer :
To determine which pair does not represent the probabilities of complementary events, we need to check if the sum of each pair equals 1. Complementary events have probabilities that add up to 1.
Let's examine each pair:
1. [tex]\(\frac{4}{7}\)[/tex] and [tex]\(\frac{3}{7}\)[/tex]:
[tex]\[ \frac{4}{7} + \frac{3}{7} = \frac{4+3}{7} = \frac{7}{7} = 1 \][/tex]
This pair sums to 1, so these are complementary events.
2. [tex]\(16 \%\)[/tex] (which can be written as 0.16) and [tex]\(84 \%\)[/tex] (which can be written as 0.84):
[tex]\[ 0.16 + 0.84 = 1 \][/tex]
This pair sums to 1, so these are complementary events.
3. 0.25 and 0.50:
[tex]\[ 0.25 + 0.50 = 0.75 \][/tex]
This pair sums to 0.75, which is not equal to 1. Hence, these are not complementary events.
4. 0 and 1:
[tex]\[ 0 + 1 = 1 \][/tex]
This pair sums to 1, so these are complementary events.
From the above calculations:
- The pairs [tex]\(\frac{4}{7}\)[/tex] and [tex]\(\frac{3}{7}\)[/tex], [tex]\(0.16\)[/tex] and [tex]\(0.84\)[/tex], and [tex]\(0\)[/tex] and [tex]\(1\)[/tex] all sum to 1 and represent complementary events.
- The pair 0.25 and 0.50 sums to 0.75, which does not represent complementary events.
Therefore, the pair that does not represent the probabilities of complementary events is:
[tex]\[ 0.25 \, \text{and} \, 0.50 \][/tex]
Let's examine each pair:
1. [tex]\(\frac{4}{7}\)[/tex] and [tex]\(\frac{3}{7}\)[/tex]:
[tex]\[ \frac{4}{7} + \frac{3}{7} = \frac{4+3}{7} = \frac{7}{7} = 1 \][/tex]
This pair sums to 1, so these are complementary events.
2. [tex]\(16 \%\)[/tex] (which can be written as 0.16) and [tex]\(84 \%\)[/tex] (which can be written as 0.84):
[tex]\[ 0.16 + 0.84 = 1 \][/tex]
This pair sums to 1, so these are complementary events.
3. 0.25 and 0.50:
[tex]\[ 0.25 + 0.50 = 0.75 \][/tex]
This pair sums to 0.75, which is not equal to 1. Hence, these are not complementary events.
4. 0 and 1:
[tex]\[ 0 + 1 = 1 \][/tex]
This pair sums to 1, so these are complementary events.
From the above calculations:
- The pairs [tex]\(\frac{4}{7}\)[/tex] and [tex]\(\frac{3}{7}\)[/tex], [tex]\(0.16\)[/tex] and [tex]\(0.84\)[/tex], and [tex]\(0\)[/tex] and [tex]\(1\)[/tex] all sum to 1 and represent complementary events.
- The pair 0.25 and 0.50 sums to 0.75, which does not represent complementary events.
Therefore, the pair that does not represent the probabilities of complementary events is:
[tex]\[ 0.25 \, \text{and} \, 0.50 \][/tex]
