Question

sarah has 52 balloons; 13 of the balloons are red, and 4 of the balloons have polka dots. just 1 of the red balloons has polka dots. what is the probability of randomly choosing a balloon that is either red or has polka dots?
a. \\(\frac{3}{13}\\)

b. \\(\frac{4}{13}\\)

c. \\(\frac{17}{32}\\)

d. \\(\frac{9}{26}\\)

Explanation:

Step1: Recall the principle of inclusion - exclusion for probability.

The formula for \( P(A \cup B) \) (probability of \( A \) or \( B \)) is \( P(A) + P(B)-P(A \cap B) \), where \( A \) is the event of choosing a red balloon and \( B \) is the event of choosing a balloon with polka dots.

Step2: Calculate \( P(A) \), \( P(B) \) and \( P(A \cap B) \).

• Total number of balloons \( n = 52 \).
• Number of red balloons \( n(A)=13 \), so \( P(A)=\frac{n(A)}{n}=\frac{13}{52} \).
• Number of balloons with polka dots \( n(B) = 4 \), so \( P(B)=\frac{n(B)}{n}=\frac{4}{52} \).
• Number of balloons that are red and have polka dots \( n(A\cap B) = 1 \), so \( P(A\cap B)=\frac{n(A\cap B)}{n}=\frac{1}{52} \).

Step3: Substitute into the inclusion - exclusion formula.

\( P(A\cup B)=\frac{13}{52}+\frac{4}{52}-\frac{1}{52}=\frac{13 + 4-1}{52}=\frac{16}{52}=\frac{4}{13}\)? Wait, no, wait: \( 13+4 - 1=16 \)? Wait, 13 + 4 is 17, 17-1 = 16? Wait, no, 13+4 = 17, 17 - 1=16? Wait, no, 13+4 is 17, 17-1 = 16? Wait, no, let's recalculate: 13 (red) + 4 (polka dots)-1 (red and polka dots) = 16? Wait, no, 13 + 4=17, 17 - 1 = 16? Wait, no, 13+4 is 17, 17-1 = 16? Wait, maybe I made a mistake. Wait, 13 red, 4 polka dots, 1 is both. So the number of balloons that are red or polka dots is \( 13 + 4-1=16 \)? Wait, no, 13 + 4=17, minus 1 (because the one that is both is counted twice) gives 16? Wait, no, 13+4 = 17, 17-1 = 16. Then \( P(A\cup B)=\frac{16}{52}=\frac{4}{13} \)? But wait, let's check the options. Wait, option D is \( \frac{9}{26}=\frac{18}{52} \), option C is \( \frac{17}{52} \), option B is \( \frac{4}{13}=\frac{16}{52} \), option A is \( \frac{3}{13} \). Wait, maybe my calculation is wrong. Wait, total balloons: 52. Red:13, polka dots:4, both:1. So number of red or polka dots is \( 13 + 4-1 = 16 \)? Wait, 13 red: 12 red non - polka, 1 red polka. Polka dots: 3 non - red polka, 1 red polka. So total red or polka: 12 + 1+3 = 16. So 16/52 = 4/13. But wait, the options: option B is 4/13. Wait, but let's re - check the problem. Wait, the problem says "Sarah has 52 balloons; 13 of the balloons are red, and 4 of the balloons have polka dots. Just 1 of the red balloons has polka dots. What is the probability of randomly choosing a balloon that is either red or has polka dots?"

Wait, maybe I made a mistake in the formula. The formula for the number of elements in \( A\cup B \) is \( n(A)+n(B)-n(A\cap B) \). So \( n(A\cup B)=13 + 4-1 = 16 \). Then probability is \( 16/52 = 4/13 \). So the answer should be B. \( \frac{4}{13} \)

Wait, but let's check again. 13 red, 4 polka dots, 1 is both. So the number of balloons that are red or polka dots is 13 + 4 - 1=16. 16 divided by 52: divide numerator and denominator by 4, we get 4/13. So option B.

Answer:

B. \(\frac{4}{13}\)