Question

In an experiment, die is rolled continually until a $6$appears, at which point the experiment stops. What is the sample space of this experiment? Let $E_n$denote the event that $n$rolls are necessary to complete the experiment. What points of the sample space are contained in $E_n$? What is${\left(\underset{1}{\overset{\infty }{{\cup E}_n}}\right)}^c$?

Short answer

The sample space contained in $E_n=5^{n-1}$

${\left(\underset{1}{\overset{\infty }{{\cup E}_n}}\right)}^c$is the event that$6$never appeared.

Step-by-step solution

Step 1. Given information

It is given that, a die is rolled continually until a 6 appears, at this point the experiment stops.

Step 2. Find the sample space and En.

If $6$appears in the first roll then the experiment stops. The possible outcome here is $6$.

Let the event that the experiment stops on the first roll of die be denoted by $E_1$and $E_1=1$.

If $6$appears in the second roll then the experiment stops. The possible outcomes here are:

role="math" localid="1648722389306" $\left\{\left(1,6\right),\left(2,6\right),\left(3,6\right),\left(4,6\right),\left(5,6\right)\right\}$

Let the event that the experiment stops on the second roll of die be denoted by $E_2$and role="math" localid="1648722693019" $E_2=5$.

If 6 appears in the third roll then the experiment stops. The possible outcomes here are:

role="math" localid="1648722679020" $$\begin{gathered} \left\{\left(1,1,6\right),\left(1,2,6\right),\left(1,3,6\right),\left(1,4,6\right),\left(1,5,6\right)\right\},\left\{\left(2,1,6\right),\left(2,2.6\right),\left(2,3,6\right),\left(2,4,6\right),\left(2,5,6\right)\right\}, \\ \left\{\left(3,1,6\right),\left(3,2,6\right),\left(3,3,6\right),\left(3,4,6\right),\left(3,5,6\right)\right\},\left\{\left(4,1,6\right),\left(4,2,6\right),\left(4,3,6\right),\left(4,4,6\right),\left(4,5,6\right)\right\}, \\ \left\{\left(5,1,6\right),\left(5,2,6\right),\left(5,3,6\right),\left(5,4,6\right),\left(5,5,6\right)\right\} \end{gathered}$$

Let the event that the experiment stops on the second roll of die be denoted by $E_3$and $E_3=25$.

Similarly, the possible no. of favorable outcomes of the happening of En is$5^{n-1}$

Step 3.  Define ∪En1∞c

$$\begin{gathered} {\left(\underset{1}{\overset{\infty }{{\cup E}_n}}\right)}^c={\left[E_1\cup E_2\cup E_3.................\right]}^c \\ ={E_1}^c\cap \hspace{0.17em}{E_2}^c\cap {E_3}^c+.............. \end{gathered}$$

= The event that $6$does not appear in the first roll, $6$does not appear in the second roll, $6$does not appear in the third roll, ...........

= The event that $6$does not appear in any roll of die.

= The event that $6$never appears.

Therefore, ${\left(\underset{1}{\overset{\infty }{{\cup E}_n}}\right)}^c$is the event that$6$never appears.