Question

FreshmenSophomoreJuniorsSeniors46Below, the two-way table is given for aclass of students.TotalMale2 2Female 36 3TotalIf a student is selected at random, find theprobability the student is a male given that it'sa senior. Round to the nearest whole percent.4[?]%

Answer 1

We are asked to find the conditional probability that the student is a male given that it's a senior.

Recall that the conditional probability is given by

`P\mleft(A\vert B\mright)=(P\mleft(A\: and\: B\mright))/(P\mleft(B\mright))`

P(A | B) means that the probability of event A given that event B has already occurred.

Applying it to the given situation,

`P(Male\vert Senior)=(P(Male\: and\: Senior))/(P(Senior))`

The probability P(Male and Senior) is given by

`P(Male\: and\: Senior)=(n(Male\: and\: Senior))/(n(total))=(2)/(30)=(1)/(15)`

Where n(Male and Senior) is the intersection of the row "Male" and the column "Senior" that is 2

n(total) is the grand total of all the students.

Grand total = 4+3+6+4+2+6+2+3 = 30

The probability P(Senior) is given by

`P(Senior)=(n(Senior))/(n(total))=(5)/(30)=(1)/(6)`

Where n(Senior) is the column total of the column "Senior" that is (2 + 3 = 5)

n(total) is the grand total of all the students.

Finally, the probability that the student is a male given that it's a senior is

`\begin{gathered} P(Male\vert Senior)=(P(Male\: and\: Senior))/(P(Senior)) \\ P(Male\vert Senior)=((1)/(15))/((1)/(6))=(1)/(15)*(6)/(1)=(6)/(15)=0.40=40\% \end{gathered}`

Therefore, the probability that the student is a male given that it's a senior is 40%