All freshmen, sophomores, juniors, and seniors attended a high school assembly. The total student attendance is shown in the table.
| Class | Number of People |
| ---------- | ---------------- |
| freshmen | 31 |
| sophomores | 10 |
| juniors | 17 |
| seniors | 22 |
Twice during the assembly, a student is chosen at random to assist with the presentation. After the first student has finished assisting, the student returns to the group and can be chosen a second time. What is the probability that the first student chosen is a senior and the second student chosen is a sophomore?
A. [tex]$\frac{11}{320}$[/tex]
B. [tex]$\frac{-3}{80}$[/tex]
C. [tex]$\frac{11}{40}$[/tex]
D. [tex]$\frac{2}{5}$[/tex]
Answer (2)
Calculate the total number of students: 31 + 10 + 17 + 22 = 80 .
Find the probability of choosing a senior first: 80 22 .
Find the probability of choosing a sophomore second: 80 10 .
Multiply the probabilities: 80 22 ⋅ 80 10 = 320 11 .
Explanation
Understand the problem and provided data Let's analyze the problem. We have a high school assembly with freshmen, sophomores, juniors, and seniors. We are given the number of people in each class. We need to find the probability that the first student chosen at random is a senior and the second student chosen at random is a sophomore, given that the first student returns to the group before the second student is chosen.
Calculate the total number of students First, we need to find the total number of students at the assembly. We add the number of students in each class: 31 + 10 + 17 + 22 = 80 . So, there are a total of 80 students.
Calculate the probability of choosing a senior first Next, we need to find the probability that the first student chosen is a senior. There are 22 seniors out of 80 total students, so the probability is 80 22 .
Calculate the probability of choosing a sophomore second Since the first student returns to the group, the total number of students remains 80. Now, we need to find the probability that the second student chosen is a sophomore. There are 10 sophomores out of 80 total students, so the probability is 80 10 .
Calculate the combined probability Since the two events are independent (the first student returns to the group), we can find the probability of both events occurring by multiplying their individual probabilities: P ( senior first and sophomore second ) = P ( senior first ) ⋅ P ( sophomore second ) = 80 22 ⋅ 80 10 = 6400 220 .
Simplify the fraction and state the final answer Now, we simplify the fraction: 6400 220 = 640 22 = 320 11 . Therefore, the probability that the first student chosen is a senior and the second student chosen is a sophomore is 320 11 .
Examples
This type of probability calculation is useful in scenarios like raffles or surveys where you want to know the likelihood of selecting specific groups of people in a particular order. For example, if a school club is randomly selecting members for different roles, they might want to calculate the probability of selecting a senior as president and a sophomore as treasurer. Understanding these probabilities helps in planning and predicting outcomes in various selection processes.
The probability that the first student chosen is a senior and the second student chosen is a sophomore is 320 11 . The calculation involves determining the total number of students and then calculating the probabilities of choosing each class in sequence. Therefore, the correct answer is option A: 320 11 .
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