To find the missing probability, [tex]\( P(A \cup B) \)[/tex], we use the formula for the union of two events. The formula is:
[tex]\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\][/tex]
Given the probabilities:
[tex]\[
P(A) = \frac{7}{20}
\][/tex]
[tex]\[
P(B) = \frac{3}{5}
\][/tex]
[tex]\[
P(A \cap B) = \frac{21}{100}
\][/tex]
Let's find the numerical values of these probabilities:
1. Converting the given fraction to a decimal, we find [tex]\( P(A) \)[/tex]:
[tex]\[
P(A) = \frac{7}{20} = 0.35
\][/tex]
2. Converting the given fraction to a decimal, we find [tex]\( P(B) \)[/tex]:
[tex]\[
P(B) = \frac{3}{5} = 0.60
\][/tex]
3. Converting the given fraction to a decimal, we find [tex]\( P(A \cap B) \)[/tex]:
[tex]\[
P(A \cap B) = \frac{21}{100} = 0.21
\][/tex]
Now, substitute these values into the formula:
[tex]\[
P(A \cup B) = 0.35 + 0.60 - 0.21
\][/tex]
Performing the addition and subtraction:
[tex]\[
P(A \cup B) = 0.95 - 0.21 = 0.74
\][/tex]
Convert the decimal result back to a fraction. Recall that:
[tex]\[
0.74 = \frac{74}{100} = \frac{37}{50}
\][/tex]
Thus, the missing probability is:
[tex]\[
P(A \cup B) = \frac{37}{50}
\][/tex]
So, the correct answer is:
[tex]\[
D. \frac{37}{50}
\][/tex]
