Question

given the data collected from 200 individuals concerning whether or not to extend the length of the school year in the table below answer the questions.

1. given that condition that a person is an adult what is the probability that they are in favor of extending the school year? p(for|adult) =
2. given the condition that a person is against extending the school year what is the probability they are a senior? p (senior|against) =
3. what is the probability that a person has no opinion given that they are a youth? p (no opinion| youth) =

Explanation:

Step1: Recall the formula for conditional probability

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In the case of frequency tables, it can be written as \(P(A|B)=\frac{\text{Frequency of }A\cap B}{\text{Frequency of }B}\)

Step13: Find \(P(\text{For}|\text{Adult})\)

• The number of adults is \(30 + 27+20=77\) (sum of "For", "Against", "No Opinion" in the "Adults" row)
• The number of adults who are "For" is \(30\)
• Using the formula \(P(\text{For}|\text{Adult})=\frac{\text{Number of adults who are For}}{\text{Number of adults}}=\frac{30}{77}\)

Step14: Find \(P(\text{Senior}|\text{Against})\)

• The number of people who are "Against" is \(35 + 27+12 = 74\) (sum of "Youth", "Adults", "Seniors" in the "Against" row)
• The number of seniors who are "Against" is \(27\)
• Using the formula \(P(\text{Senior}|\text{Against})=\frac{\text{Number of seniors who are Against}}{\text{Number of people who are Against}}=\frac{27}{74}\)

Step15: Find \(P(\text{no opinion}|\text{youth})\)

• The number of youths is \(7+35 + 12=54\) (sum of "For", "Against", "No Opinion" in the "Youth" row)
• The number of youths with "No Opinion" is \(12\)
• Using the formula \(P(\text{no opinion}|\text{youth})=\frac{\text{Number of youths with no opinion}}{\text{Number of youths}}=\frac{12}{54}=\frac{2}{9}\)

Answer:

1. \(\frac{30}{77}\)
2. \(\frac{27}{74}\)
3. \(\frac{2}{9}\)