A bug is walking a number line such that all numbers are nonnegative. The bug starts at $1$. When it lands on $0$, it goes back to $1$, but from any other number on the number line, it goes left or right with equal probability.
What is the expected number of times the bug will land on $0$ before it lands on $4$ for the first time?
2026-03-27 13:20:31.1774617631
What is the expected number of times the bug will land on $0$ before it lands on $4$ for the first time?
139 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in PROBABILITY
- How to prove $\lim_{n \rightarrow\infty} e^{-n}\sum_{k=0}^{n}\frac{n^k}{k!} = \frac{1}{2}$?
- Is this a commonly known paradox?
- What's $P(A_1\cap A_2\cap A_3\cap A_4) $?
- Prove or disprove the following inequality
- Another application of the Central Limit Theorem
- Given is $2$ dimensional random variable $(X,Y)$ with table. Determine the correlation between $X$ and $Y$
- A random point $(a,b)$ is uniformly distributed in a unit square $K=[(u,v):0<u<1,0<v<1]$
- proving Kochen-Stone lemma...
- Solution Check. (Probability)
- Interpreting stationary distribution $P_{\infty}(X,V)$ of a random process
Related Questions in EXPECTED-VALUE
- Show that $\operatorname{Cov}(X,X^2)=0$ if X is a continuous random variable with symmetric distribution around the origin
- prove that $E(Y) = 0$ if $X$ is a random variable and $Y = x- E(x)$
- Limit of the expectation in Galton-Watson-process using a Martingale
- Determine if an Estimator is Biased (Unusual Expectation Expression)
- Why are negative constants removed from variance?
- How to find $\mathbb{E}(X\mid\mathbf{1}_{X<Y})$ where $X,Y$ are i.i.d exponential variables?
- $X_1,X_2,X_3 \sim^{\text{i.i.d}} R(0,1)$. Find $E(\frac{X_1+X_2}{X_1+X_2+X_3})$
- How to calculate the conditional mean of $E(X\mid X<Y)$?
- Let X be a geometric random variable, show that $E[X(X-1)...(X-r+1)] = \frac{r!(1-p)^r}{p^r}$
- Taylor expansion of expectation in financial modelling problem
Related Questions in MARKOV-CHAINS
- Calculating probabilities using Markov chains.
- Probability being in the same state
- Random walk on $\mathbb{Z}^2$
- Polya's Urn and Conditional Independence
- Finding a mixture of 1st and 0'th order Markov models that is closest to an empirical distribution
- Find probability function of random walk, stochastic processes
- Generating cycles on a strongly connected graph
- Will be this random walk a Markov chain?
- An irreducible Markov chain cannot have an absorbing state
- Proving at least one state is recurrent in a finite-state-space Markov chain
Related Questions in RANDOM-WALK
- Random walk on $\mathbb{Z}^2$
- Density distribution of random walkers in a unit sphere with an absorbing boundary
- Monkey Random walk using binomial distribution
- Find probability function of random walk, stochastic processes
- Random walk with probability $p \neq 1$ of stepping at each $\Delta t$
- Average distance between consecutive points in a one-dimensional auto-correlated sequence
- Return probability random walk
- Random Walk: Quantiles, average and maximal walk
- motion on the surface of a 3-sphere
- Probability of symmetric random walk being in certain interval on nth step
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Shamelessly ripping offAdapting Mathmo123's method to answer the problem asked:Let $\mu_k$ denote the expected number of visits to $0$ before the first visit to $4$ if we start at $k.$ Then $$\begin{eqnarray} \mu_4 &=& 0,\\ \mu_3 &=& \frac 12 \mu_2 + \frac 12 \mu_4,\\ \mu_2 &=& \frac 12 \mu_1 + \frac 12 \mu_3,\\ \mu_1 &=& \frac 12 \mu_0 + \frac 12 \mu_2, \ \mbox{and}\\ \mu_0 &=& 1 + \mu_1.\\ \end{eqnarray}$$ The first equation is due to the fact that we can't visit anything before visiting $4$ if we start at $4;$ the last is due to the fact that being at $0$ gives us one visit, and afterward we will certainly visit $1;$ every other case is based on the $\frac 12$ probability of visiting either of the adjacent numbers next.
Solve for $\mu_1.$
A slightly more interesting question is what is the expected number of visits to $0$ before the first visit to $N$ if the bug starts at $M.$ It turns out there is a nice simple answer to that question. Apply that answer to the case $N = 4, M = 1.$