Below, a two-way table is given
for a class of students.
Male
Female
Total
Freshman Sophomore Junior
4
6
2
4
6
3
Senior
2
3
Find the probability the student is a female,
given that they are a junior.
P(female | freshman)
Total
P(female and freshman)
P(freshman)
Round to the nearest whole percent.
[?]%

The question appears to be inconsistent. In the question they are asking us to find P(female | junior) but the formula given is to find P(female | freshman).

I have computed both and you can clarify and confirm

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To find the probabilities we need to complete the totals columns and rows as well as the totals of these. The completed table is shown in the attached figure

From these we can calculate any probability

[tex]P(freshman) = \dfrac{\textrm{number of freshmen}}{\textrm{total number of students}} = \dfrac{7}{30}[/tex]

P(female and freshman) = Total number of students who are both female and freshman ÷ total number of students

Total students who are both female and freshman can be found in the cell at the intersection of row for female and column for freshman = 3

P(female and freshman) = 3/30

P(female | freshman)
[tex]= \dfrac{P(female \;and \;freshman)}{P(freshman) } = \dfrac{3/30}{7/30} \\\\= \dfrac{3}{7}\\\\\mathrm{In\;percent:}\\= \dfrac{3}{7} \times 100 %\\ \\= 42.857 \% \\\\[/tex]

Rounded to nearest percent that would be 43%

To answer the question:
Find the probability the student is a female given they are a junior

[tex]P(female\;|\;junior) = \dfrac{P(female\;and\;junior)}{P(junior)}\\\\P(female\;and\;junior) = 6/30\\\\P(junior) = \dfrac{(Number\; of \;juniors)}{30} = 8/30\\\\\dfrac{P(female\;and\;junior)}{P(junior)} = \dfrac{6/30}{8/30} = 6/8 = 3/4[/tex]

3/4 in percentage = 3/4 x 100 = 75%

So probability the student is a female given they are a junior = 75%

In fact this whole exercise would be easier to work out if you just use the numbers indicated in the columns

P(female | freshman) = number of females who are also freshman ÷ number of freshmen

= 3/7

The others can be computed the same way

semsee45 semsee45 · Answer 2 · 2024-01-23T13:53:58+01:00

Answer:

Student is female given that they are a FRESHMAN = 43%

Student is female given that they are a JUNIOR = 75%

Step-by-step explanation:

First, complete the given two-way table by adding the totals for each row and column:

[tex]\begin{array}{|l|c|c|c|c|c|}\cline{1-6}&\sf Freshman&\sf Sophomore&\sf Junior&\sf Senior&\sf Total\\\cline{1-6}\sf Male&4&6&2&2&14\\\cline{1-6}\sf Female&3&4&6&3&16\\\cline{1-6}\sf Total&7&10&8&5&30\\\cline{1-6}\end{array}[/tex]

[tex]\hrulefill[/tex]

Student is female given that they are a FRESHMAN

Reading from the two-way table:

The probability that a student is female and a freshman is 3/30.

The probability that a student is a freshman is 7/30.

Substitute this information into the given equation:

[tex]\sf P(female\;|\;freshman)=\dfrac{P(female\;and\;freshman)}{P(freshman)}[/tex]

[tex]\sf P(female\;|\;freshman)=\dfrac{\frac{3}{30}}{\frac{7}{30}}[/tex]

[tex]\sf P(female\;|\;freshman)=\dfrac{3}{7}[/tex]

[tex]\sf P(female\;|\;freshman)=0.428571428571...[/tex]

[tex]\sf P(female\;|\;freshman)=42.8571428571...\%[/tex]

[tex]\sf P(female\;|\;freshman)=43\%[/tex]

Therefore, the probability the student is female given that they are a freshman is 43%.

[tex]\hrulefill[/tex]

Student is female given that they are a JUNIOR

Reading from the two-way table:

The probability that a student is female and a junior is 6/30.

The probability that a student is a junior is 8/30.

Substitute this information into the given equation:

[tex]\sf P(female\;|\;junior)=\dfrac{P(female\;and\;junior)}{P(junior)}[/tex]

[tex]\sf P(female\;|\;junior)=\dfrac{\frac{6}{30}}{\frac{8}{30}}[/tex]

[tex]\sf P(female\;|\;junior)=\dfrac{6}{8}[/tex]

[tex]\sf P(female\;|\;junior)=0.75[/tex]

[tex]\sf P(female\;|\;junior)=75\%[/tex]

Therefore, the probability the student is female given that they are a junior is 75%.

[tex]\hrulefill[/tex]

Additional Notes

Unfortunately, there appears to be a typing error in either the question or the given formula. The question asks us to find the probability that the student is female given that they are a JUNIOR, whereas the given formula uses the word FRESHMAN instead of junior. Therefore, I have provided both calculations.