What's wrong with this sample space?

**derksheng** · 08-07-2012 10:30 AM

"A coin is tossed until for the first time the same result appears twice in succession. Describe the sample space."

My answer:

$\Omega := \{HH,TT\}\cup \{\cup_{i=1}^n \{HT_i\} \cup HH : n \in \mathbb{N}^* \} \cup \{\cup_{i=1}^n \{HT_i\} \cup T : n \in \mathbb{N}^* \}$ $\cup \{\cup_{i=1}^n \{TH_i\} \cup TT : n \in \mathbb{N}^* \} \cup \{\cup_{i=1}^n \{TH_i\} \cup H : n \in \mathbb{N}^* \}$

But the problem is that I'm getting a sample space like $\Omega := \{HH,TT,\{HH,T\},\{TT,H\}...\}$ , and I think that you're not allowed to have sets denoting individual states of the world?

**vinux** · 08-07-2012 10:57 AM

I think this is about the notation. I think the notation means
$\{\cup_{i=1}^n \{HT_i\} \cup HH = \{ HTHH, HTHTHH, HTHTHTHH,... \}$

Originally Posted by derksheng

"A coin is tossed until for the first time the same result appears twice in succession. Describe the sample space."

My answer:

$\Omega := \{HH,TT\}\cup \{\cup_{i=1}^n \{HT_i\} \cup HH : n \in \mathbb{N}^* \} \cup \{\cup_{i=1}^n \{HT_i\} \cup T : n \in \mathbb{N}^* \}$ $\cup \{\cup_{i=1}^n \{TH_i\} \cup TT : n \in \mathbb{N}^* \} \cup \{\cup_{i=1}^n \{TH_i\} \cup H : n \in \mathbb{N}^* \}$

But the problem is that I'm getting a sample space like $\Omega := \{HH,TT,\{HH,T\},\{TT,H\}...\}$ , and I think that you're not allowed to have sets denoting individual states of the world?

**derksheng** · 08-07-2012 11:32 AM

Originally Posted by vinux

I think this is about the notation. I think the notation means
$\{\cup_{i=1}^n \{HT_i\} \cup HH = \{ HTHH, HTHTHH, HTHTHTHH,... \}$

I actually made an error, it should've been $\cup_{i=1}^n \{HT_i\} \cup \{HH\}$ , instead of $\cup_{i=1}^n \{HT_i\} \cup HH$ .

Given my correction, $\cup_{i=1}^2 \{HT_i\} \cup \{HH\} = \{HT,HT,HH\} \not= \{HTHTHH\}$ (I want {HTHTHH}, and not {HT,HT,HH}.).

**vinux** · 08-07-2012 11:42 AM

My explanation was based on the solution of the problem.
$\cup_{i=1}^2 \{HT_i\} \cup \{HH\} = \{HT,HT,HH\}$ This definition doesn't help . $\{HT,HT,HH\} = \{HT,HH\}$ and HH is already in the set.
By the way what are looking for this question? solution or the expansion of the set?

Originally Posted by derksheng

I actually made an error, it should've been $\cup_{i=1}^n \{HT_i\} \cup \{HH\}$ , instead of $\cup_{i=1}^n \{HT_i\} \cup HH$ .

Given my correction, $\cup_{i=1}^2 \{HT_i\} \cup \{HH\} = \{HT,HT,HH\} \not= \{HTHTHH\}$ (I want {HTHTHH}, and not {HT,HT,HH}.).

**derksheng** · 08-07-2012 11:46 AM

Originally Posted by vinux

My explanation was based on the solution of the problem.

What solution? I posted my attempt at the solution, not that actual solution.

Originally Posted by vinux

$\cup_{i=1}^2 \{HT_i\} \cup \{HH\} = \{HT,HT,HH\}$ This definition doesn't help . $\{HT,HT,HH\} = \{HT,HH\}$ and HH is already in the set.

Great point. Didn't see this.

Originally Posted by vinux

By the way what are looking for this question? solution or the expansion of the set?

Need to find what $\Omega$ is.

**vinux** · 08-07-2012 12:12 PM

It would have been easy if I could find appropriate notation for the sequence mentioned below.

As per your question The possible outcomes are

HH, TT
HTHH, HTHTHH, HTHTHTHH, ,...
HTT, HTHTT, HTHTHT, ....
THTT, THTHTT, THTHTHTT, ...
THH, THTHH,THTHTHH,...
You need find a set notation for all the above elements.

**derksheng** · 08-07-2012 01:08 PM

Yep, thanks.

I was hoping I could find an expression for $\Omega$ where I didn't have to end it in "...", but I guess I will have to.

Thanks for your help.

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