Question

the two - way frequency table shows the results of a survey of students from each grade who plan to attend this years homecoming football game. from the table, you can see that 24 sophomores are not attending the homecoming game, and a total of 58 students are not attending the game. the table represents a sample space, with 174 total students surveyed. we can ask, \what is the probability of a sophomore not attending the homecoming game?\ this is the same as asking, \what is the probability that a randomly selected student is not attending the homecoming game, given that he or she is a sophomore?\ this can be represented by the conditional probability. to create a relative frequency table, divide each of the values in the frequency table by the total number surveyed. use the example to complete the relative frequency table. each value is rounded to the thousandth. use the relative frequency table to calculate each probability. use the frequency table in the example to verify that your answer is correct. the probability that a randomly selected student is a senior, given that they are not attending the homecoming game is approximately

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 context of a two - way frequency table, if $A$ is the event of being a sophomore and $B$ is the event of not attending the homecoming game, $P(A\cap B)$ is the number of sophomores not attending divided by the total number of students, and $P(B)$ is the number of students not attending divided by the total number of students. So $P(A|B)=\frac{\text{Number of sophomores not attending}}{\text{Number of students not attending}}$.

Step2: Identify the relevant values from the table

The number of sophomores not attending the homecoming game is 24, and the number of students not attending the homecoming game is 58.

Step3: Calculate the probability

$P=\frac{24}{58}\approx0.414$.

Answer:

The probability that a randomly selected student is a sophomore, given that the student is not attending the homecoming game is approximately $0.414$.