Question

The accompanying table shows the numbers of male and female students in a particular country who received​ bachelor's degrees in business in a recent year. Complete parts​ (a) and​ (b) below.
business degrees nonbusiness degrees total
male 189131 634650 823781
female 169539 885329 1054868
total 358670 1519979 1878649




(a) Find the probability that a randomly selected student is male​, given that the student received a business degree.
The probability that a randomly selected student is​ male, given that the student received a business​ degree, is
nothing.
​(Round to three decimal places as​ needed.)
​(b) Find the probability that a randomly selected student received a business​ degree, given that the student is female.
The probability that a randomly selected student received a business​ degree, given that the student is​ female, is
nothing.
​(Round to three decimal places as​ needed.)

Answer 1

Solution:

Probability of an event $=\frac{\text{Total favorable outcome}}{\text{Total possible outcome}}$

The table showing numbers of male and female students in a particular country who received​ bachelor's degrees in business in a recent year.

         business degrees   Non-business degrees         total

male           189131                  634650                            823781

female       169539                885329                             1054868

total           358670                 1519979                            1878649

We will use Bay's theorem to answer this question.

M=Male, F= Female, B=Business degree

(a)Probability that a randomly selected student is male​, given that the student received a business degree.

$P(\frac{M}{B})=\frac{P(\frac{B}{M})}{P(\frac{B}{M})+P(\frac{B}{F})}=\frac{\frac{189131}{358670}}{\frac{189131}{358670}+\frac{169539}{358670}}\\\\ =\frac{189131}{189131 + 169539}\\\\ =\frac{189131}{358670}\\\\ =0.527$

You can calculate it directly,

$P(\frac{M}{B})=\frac{\text{Total number of males having business degree}}{\text{Total number of business degree students}}=\frac{189131}{358670}=0.527$

(b)  The probability that a randomly selected student received a business​ degree, given that the student is Female

$P(\frac{B}{F})=\frac{\text{Girls having business degree}}{\text{Total number of girls}}\\\\ P(\frac{B}{F})=\frac{169539}{1054868}\\\\ P(\frac{B}{F})=0.161$

Answer 2

The probability of an event is the possible outcome of the event.  

• The probability that a randomly selected student is male​, given that the student received a business degree is 0.527
• The probability that a randomly selected student received a business​ degree, given that the student is female is 0.161

We make use of the following representation:

$B \to$ Business degrees

$N \to$ Non-business degrees

$M \to$ Male

$F \to$ Female

(a): The probability that a randomly selected student is male​, given that the student received a business degree.

This is represented as:

$P(M | B)$

And it is calculated as follows:

$P(M | B) = \frac{n(M\ n\ B)}{n(B)}$

From the table:

$n(M\ n\ B) = 189131$

$n(B) = 358670$

So, we have:

$P(M | B) = \frac{n(M\ n\ B)}{n(B)}$

$P(M | B) = \frac{189131}{358670 }$

$P(M | B) = 0.527$

(b): The probability that a randomly selected student received a business​ degree, given that the student is female.

This is represented as:

$P(B | F)$

And it is calculated as follows:

$P(B | F) = \frac{n(B\ n\ F)}{n(F)}$

From the table:

$n(B\ n\ F) = 169539$

$n(F) = 1054868$

So, we have:

$P(B | F) = \frac{n(B\ n\ F)}{n(F)}$

$P(B | F) = \frac{169539}{1054868}$

$P(B | F) = 0.161$

Hence,

• The probability that a randomly selected student is male​, given that the student received a business degree is 0.527
• The probability that a randomly selected student received a business​ degree, given that the student is female is 0.161

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