Chapter 13.5, Problem 24HP

To determine

To determine the value of $n$ .

Answer to Problem 24HP

There are 7 objects in the bag.

Explanation of Solution

Given information:

The probablity of drawing object $A$ and then object $B$ without replacement is 2.4%.

Probability of drawing object $A$ from $n$ object is $\frac{1}{n}$

Probability of drawing object $B$ from $n$ object without replacement is $\frac{1}{n-1}$

(This is because object $A$ is already taken so there will be $n-1$ objects left)

We know that,

P(B|A) $\cdot \text{P(A)}=\text{P(A and B)}$

Now we put values we get,

  $\begin{array}{l}0.024=\frac{1}{n}\cdot \frac{1}{n-1}\\ 0.024=\frac{1}{n^2-n}\\ n^2-n=\frac{1}{0.024}\\ n^2-n=41.66\\ n^2-n-41.66=0\end{array}$

Now we know for equation $a{x}^2+bx+c=0$ roots of equation is $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Hence,

  $\begin{array}{l}n=\frac{1\pm \sqrt{1^2-4\left(-41.66\right)}}{2}\\ n=\frac{1\pm 12.94}{2}\\ n\approx 7\end{array}$

We ignored the negative value. And we took approximate value as the number of objects is an integer value.

Hence there are 7 objects in the bag.