To determine the probability of randomly choosing a balloon that is either red or has polka dots, we need to use the formula for the union of two events:
[tex]\[ P(\text{Red or Polka Dot}) = P(\text{Red}) + P(\text{Polka Dot}) - P(\text{Red and Polka Dot}) \][/tex]
Here's a step-by-step solution to the problem:
1. Total number of balloons:
[tex]\[ 52 \][/tex]
2. Number of red balloons:
[tex]\[ 13 \][/tex]
Therefore, the probability of selecting a red balloon is:
[tex]\[ P(\text{Red}) = \frac{13}{52} = 0.25 \][/tex]
3. Number of polka dot balloons:
[tex]\[ 4 \][/tex]
Therefore, the probability of selecting a polka dot balloon is:
[tex]\[ P(\text{Polka Dot}) = \frac{4}{52} = \frac{1}{13} \approx 0.0769 \][/tex]
4. Number of balloons that are both red and have polka dots:
[tex]\[ 1 \][/tex]
Therefore, the probability of selecting a balloon that is both red and has polka dots is:
[tex]\[ P(\text{Red and Polka Dot}) = \frac{1}{52} \approx 0.0192 \][/tex]
5. Applying the formula for the union of the two events:
[tex]\[
P(\text{Red or Polka Dot}) = P(\text{Red}) + P(\text{Polka Dot}) - P(\text{Red and Polka Dot})
\][/tex]
Substituting the calculated probabilities:
[tex]\[
P(\text{Red or Polka Dot}) = 0.25 + 0.0769 - 0.0192 = 0.3077
\][/tex]
6. Converting the decimal probability back to a fraction (if required for the multiple-choice options):
[tex]\[
0.3077 \approx \frac{16}{52} = \frac{4}{13}
\][/tex]
Therefore, the probability of randomly choosing a balloon that is either red or has polka dots is [tex]\( \frac{4}{13} \)[/tex].
The correct answer is:
[tex]\[ \boxed{\frac{4}{13}} \][/tex]
