Question

compute the conditional probabilities from the two - way frequency table. drag the item from the item bank to its corresponding match. favorite summer activity for 50 children reading sports swimming total boys 2 20 8 30 girls 4 7 9 20 total 6 27 17 50 p(boy if favorite activity is swimming) p(girl if favorite activity is sports) p(girl if favorite activity is reading) p(favorite activity is reading if a girl) 0.26 0.74 0.27 0.67 0.47 0.07 0.45 0.20

Explanation:

To solve the conditional probability problems, we use the formula for conditional probability: \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), which for frequency tables translates to \( P(A|B) = \frac{\text{Frequency of } A \cap B}{\text{Frequency of } B} \).

1. \( P(\text{Boy} | \text{favorite activity is swimming}) \)

Step 1: Identify the frequencies

• Frequency of (Boy and Swimming) = 8 (from the table, Boys who like Swimming)
• Frequency of (Swimming) = 17 (Total who like Swimming)

Step 2: Apply the conditional probability formula

\( P(\text{Boy} | \text{Swimming}) = \frac{8}{17} \approx 0.47 \)

2. \( P(\text{Girl} | \text{favorite activity is sports}) \)

Step 1: Identify the frequencies

• Frequency of (Girl and Sports) = 7 (from the table, Girls who like Sports)
• Frequency of (Sports) = 27 (Total who like Sports)

Step 2: Apply the conditional probability formula

\( P(\text{Girl} | \text{Sports}) = \frac{7}{27} \approx 0.26 \)

3. \( P(\text{Girl} | \text{favorite activity is reading}) \)

Step 1: Identify the frequencies

• Frequency of (Girl and Reading) = 4 (from the table, Girls who like Reading)
• Frequency of (Reading) = 6 (Total who like Reading)

Step 2: Apply the conditional probability formula

\( P(\text{Girl} | \text{Reading}) = \frac{4}{6} \approx 0.67 \)

4. \( P(\text{Favorite activity is reading} | \text{a girl}) \)

Step 1: Identify the frequencies

• Frequency of (Reading and Girl) = 4 (from the table, Girls who like Reading)
• Frequency of (Girl) = 20 (Total number of Girls)

Step 2: Apply the conditional probability formula

\( P(\text{Reading} | \text{Girl}) = \frac{4}{20} = 0.20 \)

Final Matches:

• \( P(\text{Boy} | \text{favorite activity is swimming}) \approx 0.47 \)
• \( P(\text{Girl} | \text{favorite activity is sports}) \approx 0.26 \)
• \( P(\text{Girl} | \text{favorite activity is reading}) \approx 0.67 \)
• \( P(\text{Favorite activity is reading} | \text{a girl}) = 0.20 \)

So the matches are:

• \( P(\text{Boy} | \text{favorite activity is swimming}) \) → 0.47
• \( P(\text{Girl} | \text{favorite activity is sports}) \) → 0.26
• \( P(\text{Girl} | \text{favorite activity is reading}) \) → 0.67
• \( P(\text{Favorite activity is reading} | \text{a girl}) \) → 0.20

Answer:

To solve the conditional probability problems, we use the formula for conditional probability: \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), which for frequency tables translates to \( P(A|B) = \frac{\text{Frequency of } A \cap B}{\text{Frequency of } B} \).

1. \( P(\text{Boy} | \text{favorite activity is swimming}) \)

Step 1: Identify the frequencies

• Frequency of (Boy and Swimming) = 8 (from the table, Boys who like Swimming)
• Frequency of (Swimming) = 17 (Total who like Swimming)

Step 2: Apply the conditional probability formula

\( P(\text{Boy} | \text{Swimming}) = \frac{8}{17} \approx 0.47 \)

2. \( P(\text{Girl} | \text{favorite activity is sports}) \)

Step 1: Identify the frequencies

• Frequency of (Girl and Sports) = 7 (from the table, Girls who like Sports)
• Frequency of (Sports) = 27 (Total who like Sports)

Step 2: Apply the conditional probability formula

\( P(\text{Girl} | \text{Sports}) = \frac{7}{27} \approx 0.26 \)

3. \( P(\text{Girl} | \text{favorite activity is reading}) \)

Step 1: Identify the frequencies

• Frequency of (Girl and Reading) = 4 (from the table, Girls who like Reading)
• Frequency of (Reading) = 6 (Total who like Reading)

Step 2: Apply the conditional probability formula

\( P(\text{Girl} | \text{Reading}) = \frac{4}{6} \approx 0.67 \)

4. \( P(\text{Favorite activity is reading} | \text{a girl}) \)

Step 1: Identify the frequencies

• Frequency of (Reading and Girl) = 4 (from the table, Girls who like Reading)
• Frequency of (Girl) = 20 (Total number of Girls)

Step 2: Apply the conditional probability formula

\( P(\text{Reading} | \text{Girl}) = \frac{4}{20} = 0.20 \)

Final Matches:

• \( P(\text{Boy} | \text{favorite activity is swimming}) \approx 0.47 \)
• \( P(\text{Girl} | \text{favorite activity is sports}) \approx 0.26 \)
• \( P(\text{Girl} | \text{favorite activity is reading}) \approx 0.67 \)
• \( P(\text{Favorite activity is reading} | \text{a girl}) = 0.20 \)

So the matches are:

• \( P(\text{Boy} | \text{favorite activity is swimming}) \) → 0.47
• \( P(\text{Girl} | \text{favorite activity is sports}) \) → 0.26
• \( P(\text{Girl} | \text{favorite activity is reading}) \) → 0.67
• \( P(\text{Favorite activity is reading} | \text{a girl}) \) → 0.20