Chapter 4.3, Problem 20E

(a)

To determine

To find:

The probability that a sophomore boy and then a freshman girl are chosen.

Answer to Problem 20E

Solution:

The probability of the given event is 0.03019.

Explanation of Solution

Given information:

Mrs. Harvey’s algebra class has 42 students, classified by academic year and gender as follows.

Mrs. Harvey’s Algebra Class
	Male	Female
Freshman	9	13
Sophomore	4	5
Junior	4	2
Senior	2	3

Definition:

Dependent Event:

Two events are dependent if one event happening effect the probability of the other event happening.

Formula used:

If $E\text{and}F$ are two dependent events, then the formula to calculate the probability that $E\text{and}F$ occur, is,

$\begin{array}{rll}P\left(E\text{and}F\right)& =& P\left(E\right)\cdot P\left(F/E\right)\\ & =& P\left(F\right)\cdot P\left(E/F\right)\end{array}$

$P\left(F/E\right)$ denotes the conditional probability of $F$, given that $E$ has already occurred.

Calculation:

Total number of student is 42. Total number of student will be reduced by 1, in 2^nd selection.

There are 4 boy sophomores.

There are 13 girl freshmen.

Let $A$ be the event of selecting a sophomore boy and $B$ be the event of selecting a freshman girl.

$\raisebox{1ex}{$B$}\!\left/ \!\raisebox{-1ex}{$A$}\right.$ be the event of selecting a freshman girl after selecting a sophomore boy.

Calculate the probability of event $A$.

$P\left(A\right)=\frac{4}{42}$

Calculate the probability of event $\raisebox{1ex}{$B$}\!\left/ \!\raisebox{-1ex}{$A$}\right.$.

$\begin{array}{rll}P\left(\raisebox{1ex}{$B$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)& =& \frac{13}{42-1}\\ & =& \frac{13}{41}\end{array}$

$A\text{and}B$ are dependent events.

Use the multiplication rule to calculate the probability of dependent events $A\text{and}B$.

$\begin{array}{rll}P\left(A\text{and}B\right)& =& P\left(A\right)\cdot P\left(B/A\right)\\ & =& \frac{4}{42}\times \frac{13}{41}\\ & =& \frac{52}{1722}\\ & =& 0.03019\end{array}$

Conclusion:

Thus, the probability of the given event is 0.03019.

(b)

To determine

To find:

The probability that two boys are chosen.

Answer to Problem 20E

Solution:

The required probability is 0.1986.

Explanation of Solution

Given information:

Mrs. Harvey’s algebra class has 42 students, classified by academic year and gender as follows.

Mrs. Harvey’s Algebra Class
	Male	Female
Freshman	9	13
Sophomore	4	5
Junior	4	2
Senior	2	3

Definition:

Dependent Event:

Two events are dependent if one event happening effect the probability of the other event happening.

Formula used:

If $E\text{and}F$ are two dependent events, then the formula to calculate the probability that $E\text{and}F$ occur, is,

$\begin{array}{rll}P\left(E\text{and}F\right)& =& P\left(E\right)\cdot P\left(F/E\right)\\ & =& P\left(F\right)\cdot P\left(E/F\right)\end{array}$

$P\left(F/E\right)$ denotes the conditional probability of $F$, given that $E$ has already occurred.

Calculation:

Total number of student is 42. Total number of student will be reduced by 1, in 2^nd selection.

Total number of boys is $\left(9+4+4+2\right)=19$. Total number of boy will be reduced by 1, in 2^nd selection.

Let $A$ be the event of selecting a boy and $B$ be the event of selecting another boy.

$\raisebox{1ex}{$B$}\!\left/ \!\raisebox{-1ex}{$A$}\right.$ be the event of selecting a boy after selecting a boy.

Calculate the probability of event $A$.

$P\left(A\right)=\frac{19}{42}$

Calculate the probability of event $\raisebox{1ex}{$B$}\!\left/ \!\raisebox{-1ex}{$A$}\right.$.

$\begin{array}{rll}P\left(\raisebox{1ex}{$B$}\!\left/ \!\raisebox{-1ex}{$A$}\right.\right)& =& \frac{19-1}{42-1}\\ & =& \frac{18}{41}\end{array}$

$A\text{and}B$ are dependent events.

Use the multiplication rule to calculate the probability of dependent events $A\text{and}B$.

$\begin{array}{rll}P\left(A\text{and}B\right)& =& P\left(A\right)\cdot P\left(B/A\right)\\ & =& \frac{19}{42}\times \frac{18}{41}\\ & =& \frac{342}{1722}\\ & =& 0.1986\end{array}$

Conclusion:

Thus, the required probability is 0.1986.