Question

The experiment is to repeatedly toss a coin until first tail shows up. Identify the type of the sample space.

• A. Finite sample space
• B. Continuous sample space
• C. Infinite discrete sample space
• D. None of these

Answer 1

The correct option is C Infinite discrete sample space

The experiment is to repeatedly toss a coin until first tail shows up.

A coin is tossed. The possible outcomes are $head-H,tail-T$.

Look into the following table:

$1^{st}$ toss	$2^{nd}$ toss	$3^{rd}$ toss	$4^{th}$ toss	$5^{th}$ toss
$T$	$-$	$-$	$-$	$-$
$H$	$T$	$-$	$-$	$-$
$H$	$H$	$T$	$-$	$-$
$H$	$H$	$H$	$T$	$-$
$H$	$H$	$H$	$H$	$T$

If we get $T$ at first toss, then our experiment ends.

Otherwise second toss. If we get $T$, out experiment ends. If not the process continues till we end up in tail.

Hence the possible outcomes are sequences of $H$ that, if finite, end with a single $T$, and an infinite sequence of $H$.

Therefore, $S=\{T,HT,HHT,HHHT,HHHHT,...,\{HHHH....\left\}\right\}$

Thus we get a infinite but countable(depends on the number of toss) sample space.

As we shall see elsewhere, this is a remarkable space that contains a not impossible event whose probability is $0$.

We know that, "a discrete sample space is one with a finite or countably infinite number of possible values."

Hence, it is a infinite discrete sample space.