For this problem, assume that three students are chosen from 3 freshmen, 7 sophomores, and 5 juniors. What is the probability of selecting 2 freshmen and 1 junior given that at least one freshman will be selected?

For this problem, assume that three students are chosen from 3 freshmen, 7 sophomores, and 5 juniors.
What is the probability of selecting 2 freshmen and 1 junior given that at least one freshman will be selected?
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

falhiblesw
Step 1
Given: Number of Freshmen, Sophomores & juniors are 3, 7 & 5 respectively.
To find: Probability of selecting 2 freshmen, 1 junior given that at least one freshman will be selected.
Total number of students is equal to the sum of number of freshmen, sophomores and juniorsTotal number of students $3+7+5$
$=15$ The formula which gives us the total number of ways of forming group of r objects, out of n objects is:
$C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)}$
Step 2
Three students are selected from 3 freshmen, 7 sophomores and 5 juniors, Such that the at least 1 freshman is selected
The selection of one freshman can be done in ways
$P\left(3,1\right)=\frac{3!}{\left(3-1\right)!}=3$
Selection of one freshman can be done in 3 ways
The selection of remaining two students from 2 freshmen, 7 sophomores and 5 juniors.
Number of ways of selection two students out of $\left(2+7+5\right)=14$ students
$C\left(14,2\right)=\frac{14!}{2!\left(14-2\right)!}$
$C\left(14,2\right)=7×13$
$C\left(14,2\right)=91$
Thus total number of possible ways of selection three students are $=3×91=273$
Step 3
Selection of two freshman from three freshmen is done in ways
$C\left(3,2\right)=\frac{3!}{2!\left(3-2\right)!}$

Thus, the selection of two freshman and one junior can be done in ways

The selection of two freshman and one junior can be done in 9 ways.
Step 4
Probability of selection of two freshman and one junior out of the total number of possible ways of selection three students such that at least one freshman is selected.
Number of favorable outcomes $=9$
Total number of outcomes $=273$
Probability $=\frac{\text{Number of favorable outcomes}}{\text{total number of outcomes}}$
Probability $=\frac{9}{273}$
Probability $=0.032$
Probability of selection of two freshman and one junior given that at least one freshman is selected is 0.032.
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 15, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://plainmath.net/6642/students-freshmen-sophomores-probability-selecting-freshmen-freshman", "fetch_time": 1656586653000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2022-27/segments/1656103671290.43/warc/CC-MAIN-20220630092604-20220630122604-00011.warc.gz", "warc_record_offset": 496629263, "warc_record_length": 13657, "token_count": 684, "char_count": 2500, "score": 4.625, "int_score": 5, "crawl": "CC-MAIN-2022-27", "snapshot_type": "latest", "language": "en", "language_score": 0.8901462554931641, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00023-of-00064.parquet"}