To solve for the probability [tex]\( P(A \cup B) \)[/tex], we can use the formula for the probability of the union of two events. The formula for the union of two events can be written as:
[tex]\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \][/tex]
We are given:
[tex]\[ P(A) = \frac{7}{20} \][/tex]
[tex]\[ P(B) = \frac{3}{5} \][/tex]
[tex]\[ P(A \cap B) = \frac{21}{100} \][/tex]
First, let's convert all fractions to a common denominator to simplify the calculations. The common denominator for 20, 5, and 100 is 100.
Convert [tex]\( P(A) \)[/tex]:
[tex]\[ P(A) = \frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100} \][/tex]
Convert [tex]\( P(B) \)[/tex]:
[tex]\[ P(B) = \frac{3}{5} = \frac{3 \times 20}{5 \times 20} = \frac{60}{100} \][/tex]
Now we can apply the formula:
[tex]\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \][/tex]
Substitute the values:
[tex]\[ P(A \cup B) = \frac{35}{100} + \frac{60}{100} - \frac{21}{100} \][/tex]
Now, let's perform the addition and subtraction:
[tex]\[ P(A \cup B) = \frac{35 + 60 - 21}{100} = \frac{74}{100} \][/tex]
Finally, let's simplify this fraction if possible. In this case, [tex]\(\frac{74}{100}\)[/tex] simplifies to [tex]\(\frac{37}{50}\)[/tex].
Therefore, the missing probability [tex]\( P(A \cup B) \)[/tex] is:
[tex]\[ \boxed{\frac{37}{50}} \][/tex]
So the correct answer is:
B. [tex]\(\frac{37}{50}\)[/tex]
