Question

	for	against	no opinion	total
youth (5 to 19)	7	35	12
adults (20 to 55)	30	27	20
seniors (55+)	25	16	28
total				200

1. given the condition that a person is an adult, what is the probability that they are in favor of extending the school year?

$p(\text{for}|\text{adult}) = $

1. given the condition that a person is against extending the school year, what is the probability they are a senior?

$p(\text{senior}|\text{against}) = $

1. what is the probability that a person has no opinion given that they are a youth?

$p(\text{no opinion}|\text{youth}) = $

Explanation:

Step1: Calculate Total for Adults

Sum the values in the Adults row: \( 30 + 27 + 20 = 77 \)? Wait, no, wait. Wait, first, let's fill the table properly. Wait, the total is 200. Let's check Youth total: \( 7 + 35 + 12 = 54 \). Adults: \( 30 + 27 + 20 = 77 \). Seniors: \( 25 + 16 + 28 = 69 \). Wait, 54 + 77 + 69 = 200? 54+77=131, 131+69=200. Yes. Now, for question 13: \( P(\text{For} | \text{Adult}) = \frac{\text{Adults For}}{\text{Total Adults}} \). Adults For is 30, Total Adults is 77? Wait, no, wait: Adults row total is 30 + 27 + 20 = 77. So \( P(\text{For} | \text{Adult}) = \frac{30}{77} \approx 0.3896 \). Wait, but let's confirm.

Step2: Question 14: \( P(\text{Senior} | \text{Against}) \)

First, find total Against: Youth Against 35, Adults Against 27, Seniors Against 16. So total Against: \( 35 + 27 + 16 = 78 \). Seniors Against is 16. So \( P(\text{Senior} | \text{Against}) = \frac{16}{78} = \frac{8}{39} \approx 0.2051 \).

Step3: Question 15: \( P(\text{No Opinion} | \text{Youth}) \)

Youth No Opinion is 12, Total Youth is \( 7 + 35 + 12 = 54 \). So \( P(\text{No Opinion} | \text{Youth}) = \frac{12}{54} = \frac{2}{9} \approx 0.2222 \).

Wait, let's recheck the table:

Youth (5-19): For=7, Against=35, No Opinion=12, Total=7+35+12=54.

Adults (20-55): For=30, Against=27, No Opinion=20, Total=30+27+20=77.

Seniors (55+): For=25, Against=16, No Opinion=28, Total=25+16+28=69.

Total For: 7+30+25=62.

Total Against: 35+27+16=78.

Total No Opinion: 12+20+28=60.

Total: 62+78+60=200. Correct.

Now, question 13: \( P(\text{For} | \text{Adult}) = \frac{\text{Adults For}}{\text{Total Adults}} = \frac{30}{77} \approx 0.3896 \).

Question 14: \( P(\text{Senior} | \text{Against}) = \frac{\text{Seniors Against}}{\text{Total Against}} = \frac{16}{78} = \frac{8}{39} \approx 0.2051 \).

Question 15: \( P(\text{No Opinion} | \text{Youth}) = \frac{\text{Youth No Opinion}}{\text{Total Youth}} = \frac{12}{54} = \frac{2}{9} \approx 0.2222 \).

Answer:

1. \( \frac{30}{77} \) (or approximately 0.39)
2. \( \frac{8}{39} \) (or approximately 0.21)
3. \( \frac{2}{9} \) (or approximately 0.22)