Intereting Posts

Localising a polynomial ring and non-maximal prime ideal
Equivalence relations and equivalence classes
A dilogarithm identity?
If $n-1$ is $f(n)$-smooth, $n$ is prime infinitely often. What is the best $f$?
ODE $d^2y/dx^2 + y/a^2 = u(x)$
Is half-filled magic square problem NP-complete?
Find the sum $\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + …+ \frac{1}{\sqrt{99}+\sqrt{100}}$
How to find a general sum formula for the series: 5+55+555+5555+…?
When is the category of (quasi-coherent) sheaves of finite homological dimension?
Legendre transform of a norm
Prove weak convergence of a sequence of discrete random variables
What is meant by parameter in this context?
Connectedness of a regular graph and the multiplicity of its eigenvalue
Counting the exact number of coin tosses
Inverse Function Theroem in $R^1$

In the spirit of this question where we calculated the probability of at least once having gotten at least $k$ heads after $n$ coin tosses.

How would we go about calculating the probability of having gotten at least $k$ heads in a row $l$ times over after $m$ flips. What ideas or techniques can we utilize to make it easy to generalize this? I am more interested in description of the tools and techniques one can use than the actual answer.

**Examples:**

- Differences of consecutive hitting times
- Period of a Markov Chain: Why is this one aperiodic?
- Good introductory book for Markov processes
- Sum of i.i.d. random variables is a markov chain
- Transformation of state-space that preserves Markov property
- When the sum of independent Markov chains is a Markov chain?

2 times 2 heads in a row

$HTH{\bf H}TTTH{\bf H} \leftarrow$ first time becomes true here.

2 times 3 heads in a row:

$HTHH{\bf H}THHTTHH{\bf H} \leftarrow$ first time becomes true here.

- Probability of crossing a point in a given time window
- Memoryless property of the exponential distribution
- Bounds for the maximum of binomial random variables
- Find the population size that maximizes the probability that two random samples of size $20$ will have exactly $2$ members in common
- What's wrong with my solution for the birthday problem?
- Can conditional distributions determine the joint distribution?
- Is $g(u)= \frac{E }{E }$ decreasing in $u$
- finding Expected Value for a system with N events all having exponential distribution
- Probability of ball ownership
- Non-centered Gaussian moments

Let’s take a look at a minimal example:

$$P = \frac 1 2\left[\begin{array}{ccc|ccc}

2&1&0&0&0&0\\

0&0&1&0&0&0\\

0&1&1&\color{red} 2&0&0\\\hline

0&0&0&\color{red} \uparrow&1&0\\

0&0&0&0&0&1\\

0&0&0&0&1&1

\end{array}\right]$$

- Basic “building blocks” as in the answer here on the “diagonal” we can easily implement with Kronecker product with $\bf I$, the “storage states” are the 2s.
- Now let but the upper leftmost block have storage state displaced 1 row to the block above.
- This way each displacement will slow down (1 lower exponent than the previous part of the chain).

**Any elegant way to avoid the displacement latency will be welcome!**

Crazy checker (first time we get non-zero prob for 1 and 2 resp):

$${\bf v} = \left[\begin{array}{cccccc}0&0&0&0&0&1\end{array}\right]^T$$

$${\bf P}^2{\bf v} = \left[\begin{array}{cccccc}

0&0&0&0.25&0.25&0.5\end{array}\right]^T\\{\bf P}^5{\bf v} = \left[\begin{array}{cccccc}0.0625&0.125&0.3125&0.09375&0.15625&0.25\end{array}\right]^T$$

The chance for 2 heads in a row is $1/4 = 0.25$

The chance for 4 heads in a row is $1/16 = 0.0625$

So assuming $H{\bf H}H{\bf H}$ counts as two heads in a row twice then the solution works!

We can also calculate the probability of not having any two H in a row in a string of 2 and 5 respectively, and if we do, we do indeed get the sum of the two last states: $$1-\frac 1 4\approx 0.25+0.5 = 0.75 \\ 1-\frac {19}{32} \approx 0.15625+0.25= 0.40625$$

The systematic construction of matrices for arbitrary $k,l,m$ should now be obvious.

- Must an ideal contain the kernel for its image to be an ideal?
- Prove $A^tB^t = (BA)^t$
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- mapping properties of $(1−z)^i$
- Consistency of matrix norm: $||Ax||_2 \leq ||A||_{Frobenius}||x||_2$
- Need help understanding statement of Van Kampen's Theorem and using it to compute the fundamental group of Projective Plane
- Show the Dedekind $\eta(\tau)$ function is a modular form of weight $\frac{1}{2}$ for $\Gamma_0(6)$
- Counting the number of polygons in a figure
- Prove that 16, 1156, 111556, 11115556, 1111155556… are squares.
- Characterize the commutative rings with trivial group of units
- Infinite tetration, convergence radius
- How to deal with polynomial quotient rings
- Why does $A^TA=I, \det A=1$ mean $A$ is a rotation matrix?
- Show that the quotient group $T/N$ is abelian
- Does the proof of Bolzano-Weierstrass theorem require axiom of choice?