Q.4.1 - A First Course in Probability

Question

Two balls are chosen randomly from an urn containing$8$white,$4$ black, and$2$ orange balls. Suppose that we win $\backslash (2$for each black ball selected and we lose $\backslash )1$ for each white ball selected. Let $X$ denote our winnings. What are the possible values of $X$, and what are the probabilities associated with each value?

Step-by-Step Solution

Answer

Possible values of $\mathrm{X}:-2,-1,0,1,2,4$

Probabilities associated with succeeding values $(-2,-1,0,1,2,4)\text{ are }\frac{28}{91},\frac{16}{91},\frac{1}{91},\frac{32}{91},\frac{8}{91},\frac{6}{91}$ respectively.

Step 1:Given Information

There are$8\text{ white, }4\text{ black, and }2$ orange balls in the urn. In the

 circumstance of pulling two balls at random. Random variable $\text{ X }$ represents the winning amount. The values of $\text{ X corresponding }$ to the results of the event are as documented.

WW,WO,OO,WB,BO,BB

Outcomes	WW	WO	OO	WB	BO	BB
$X$	$-2$	$-1$	$0$	$1$	$2$	$4$

Step 2:Explanation

The total number of paths of selecting $2$balls from the urn holding $8+4+2$ balls is $14C2$. The number of paths of selecting two white balls is $8C2$ as there are only $8$ white balls.

$\mathrm{X}:-2,-1,0,1,2,4$

$P(X=-2)=\frac{n_{WW}}{n_{\text{total }}}=\frac{\left(\begin{array}{c}8\\ 2\end{array}\right)}{\left(\begin{array}{c}14\\ 2\end{array}\right)}=\frac{28}{91}$

Step 3:Chances of selecting WO

The number of routes of selecting one white and one orange ball is  as both are independent occurrences

$P(X=-1)=\frac{n_{WO}}{n_{\text{total }}}=\frac{\left(\begin{array}{c}8\\ 1\end{array}\right)\left(\begin{array}{c}2\\ 1\end{array}\right)}{\left(\begin{array}{c}14\\ 2\end{array}\right)}=\frac{16}{91}$

Step 4:Chances of selecting OO

The number of routes of selecting both orange balls

$P(X=0)=\frac{n_{OO}}{n_{\text{total }}}=\frac{\left(\begin{array}{c}2\\ 2\end{array}\right)}{\left(\begin{array}{c}14\\ 2\end{array}\right)}=\frac{1}{91}$

Step 5:Chances of selecting WB

The number of routes of selecting one white and one black balls

$P(X=1)=\frac{n_{WB}}{n_{\text{total }}}=\frac{\left(\begin{array}{l}8\\ 1\end{array}\right)\left(\begin{array}{l}4\\ 1\end{array}\right)}{\left(\begin{array}{c}14\\ 2\end{array}\right)}=\frac{32}{91}$

Step 6:Chances of selecting BO

The number of routes of selecting one black and one orange ball

$P(X=2)=\frac{n_{OB}}{n_{\text{total }}}=\frac{\left(\begin{array}{c}2\\ 1\end{array}\right)\left(\begin{array}{l}4\\ 1\end{array}\right)}{\left(\begin{array}{c}14\\ 2\end{array}\right)}=\frac{8}{91}$

Step 7:Chances of selecting BB

The number of routes of selecting both black balls

$P(X=4)=\frac{n_{BB}}{n_{\text{total }}}=\frac{\left(\begin{array}{l}4\\ 2\end{array}\right)}{\left(\begin{array}{c}14\\ 2\end{array}\right)}=\frac{6}{91}$

Step 8:Final Answer

Probabilities associated with succeeding values$(-2,-1,0,1,2,4)\text{ are }\frac{28}{91},\frac{16}{91},\frac{1}{91},\frac{32}{91},\frac{8}{91},\frac{6}{91}$ respectively.