Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Toss a coin three times, so event space $\Omega=\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}$. We win $\$1$ if we flip a Head and lose $\$1$ for a Tail. Let $\mathbb{P}(H) = p$ and $\mathbb{P}(T) = q$. The change in our wealth after flip $i$ is the r.v.

$$X_i = \cases{+1 \text{ if }H \\ -1 \text{ if } T}$$

Our wealth after turn $i$ is:

$$ S_i = S_0 + X_1 + \dots X_i $$

There are three questions (these are not homework problems, rather revision) and I have some questions about their solutions.

  1. Find $\mathbb{P}(S_3|S_1)$

This is the probability of $S_3$ occuring given that $S_1$ occurs, and by definition:

$$ \mathbb{P}(S_3|S_1) = \frac{\mathbb{P}(S_1 \cap S_3)}{\mathbb{P}(S_1)} $$

I think the answer should be $\frac{1}{2}$ thinking of the $H/T$ outcome as paths on a binary tree. Can someone provide a more algebraic solution?

  1. Find $\mathbb{E}(S_3|S_1)$

We interpret this as our expected wealth after $3$ flips given $S_1$. This is just: $S_1 + p^2(2) + 2pq(0) - q^2(2) = S_1$ iff $p=q$. I think this is okay.

  1. Given that $F_0 = \{\phi,\Omega\}$ what are $F_1, F_2$ and $F_3$ where $F_i$ is the smallest event space that we can identify from complete knowledge of $F_j$ for $1\leq j\leq i$.

I am unfamiliar with filtrations, and have only a vague sense of what they actually are. So is $F_1$ is the event space we can identify from complete knowledge of the first coin flip? What does it mean to identify an event space from complete knowledge?

share|cite|improve this question
The meaning of the phrase the probability of $S_3$ occuring given that $S_1$ occurs is unclear since $S_i$ are random variables, not events. – Did Jun 19 '12 at 7:29
So there is no single value we can associate with $P(S_3|S_1)$ in this case? – anthus Jun 19 '12 at 8:14
Personally, I never use expressions like $P(S_3\mid S_1)$ and I do not know what they could mean. In your view, what could be a definition of $P(S_3\mid S_1)$ as a real number? – Did Jun 19 '12 at 8:34
up vote 2 down vote accepted

$S_1$ and $S_3$ are random variables rather than events, so when reading $\Pr(S_3|S_1)$ in your first question you may want to say something like $$\Pr(S_3=s+2|S_1=s) = \frac14$$ $$\Pr(S_3=s|S_1=s) = \frac12$$ $$\Pr(S_3=s-2|S_1=s) = \frac14.$$

share|cite|improve this answer
Thanks Henry, this is much clearer. Could you comment on the filtration, $F = (F_1,F_2,F_3)$? What is $F_1$ for instance. I know that $F_1 \subset F_2$ and that these are subsets of the event space. But the meaning of $F_i$ is not clear to me. – anthus Jun 19 '12 at 8:03
$\mathcal{F_1}$ will be the sigma algebra based on $\{HHH,HHT,HTH,HTT\}$ and $\{THH,THT,TTH,TTT\}$ since you can only distinguish the first event, while $\mathcal{F_2}$ will be the sigma algebra based on $\{HHH,HHT\},$ $\{HTH,HTT\}$, $\{THH,THT\}$ and $\{TTH,TTT\}$, and $\mathcal{F_3}$ will be the power set of your $\Omega$.… might help – Henry Jun 19 '12 at 21:23

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.