Recent questions in Geometric Probability

High school geometryAnswered question

Theresa Daugherty 2022-12-06

The degree measure of a semi-circle is

1) ${0}^{\circ}$

2) ${90}^{\circ}$

3) ${360}^{\circ}$

4) ${180}^{\circ}$

1) ${0}^{\circ}$

2) ${90}^{\circ}$

3) ${360}^{\circ}$

4) ${180}^{\circ}$

High school geometryAnswered question

Sophie Marks 2022-11-21

Geometric Probability Problem, Random Numbers $0-1+$ Triangles.

Randy presses RANDOM on his calculator twice to obtain two random numbers between 0 and 1. Let p be the probability that these two numbers and 1 form the sides of an obtuse triangle. Find p.

Randy presses RANDOM on his calculator twice to obtain two random numbers between 0 and 1. Let p be the probability that these two numbers and 1 form the sides of an obtuse triangle. Find p.

High school geometryAnswered question

Ty Moore 2022-11-17

Negative Binomial Vs Geometric

So I am trying to get the difference between these two distributions. I think I understand them but the negative binomial has me a bit confused.

The Geometric is the probability of some amount of successes before the first failure.

The Negative binomial is the portability of some amount of successes before a specified number of failures? for example the number of successes before the 8th failure?

So I am trying to get the difference between these two distributions. I think I understand them but the negative binomial has me a bit confused.

The Geometric is the probability of some amount of successes before the first failure.

The Negative binomial is the portability of some amount of successes before a specified number of failures? for example the number of successes before the 8th failure?

High school geometryAnswered question

Kale Sampson 2022-11-17

Investigating the generalised Hamming distance of some code and information transmission of a binary code.

Letting A be an alphabet of size a and $S={A}^{n}$. Meaning the set of all worlds of length n with bits chosen from the alphabet A. We now take $w\in S$.

Assume $n=14$. How many words in S are of Hamming distance less than or equal to 4 from w?

Secondly, assume $n=14$ and $a=2$ and that $w\in S$ is transmitted through a Binary Symmetric Channel with probability p of correct transmission for each individual bit. How would you generalise a formula for the probability that 5 or fewer errors will occur during transmission of w?

Letting A be an alphabet of size a and $S={A}^{n}$. Meaning the set of all worlds of length n with bits chosen from the alphabet A. We now take $w\in S$.

Assume $n=14$. How many words in S are of Hamming distance less than or equal to 4 from w?

Secondly, assume $n=14$ and $a=2$ and that $w\in S$ is transmitted through a Binary Symmetric Channel with probability p of correct transmission for each individual bit. How would you generalise a formula for the probability that 5 or fewer errors will occur during transmission of w?

High school geometryAnswered question

spasiocuo43 2022-11-15

approximate probability of geometric distribution using CLT

For $i\ge 1$, let ${X}_{i}\sim {G}_{1}/2$ be distributed Geometrically with parameter 1/2. Define ${Y}_{n}=\frac{1}{\sqrt{n}}\sum _{i=1}^{n}({X}_{i}-2)$

Approximate $P(-1\le {Y}_{n}\le 2)$ with large enough n. Hint, note that ${Y}_{n}$ is not "properly" normalized.

For $i\ge 1$, let ${X}_{i}\sim {G}_{1}/2$ be distributed Geometrically with parameter 1/2. Define ${Y}_{n}=\frac{1}{\sqrt{n}}\sum _{i=1}^{n}({X}_{i}-2)$

Approximate $P(-1\le {Y}_{n}\le 2)$ with large enough n. Hint, note that ${Y}_{n}$ is not "properly" normalized.

High school geometryAnswered question

Nola Aguilar 2022-11-15

Negative binomial distribution vs geometric distributiion

How can this question be geometric and also negative binomial. In order to join a group Peter needs 9 invitations.The probability that he receives an invitation on any day is 0.8, independent of other days.

He joins as soon as he receives his 9 th invitation. Given that he joined on the 14 th day, find the probability that he receives his first invite on the first day.

How can this question be geometric and also negative binomial. In order to join a group Peter needs 9 invitations.The probability that he receives an invitation on any day is 0.8, independent of other days.

He joins as soon as he receives his 9 th invitation. Given that he joined on the 14 th day, find the probability that he receives his first invite on the first day.

High school geometryAnswered question

piopiopioirp 2022-11-14

What is the probability that the quadratic equation $a{x}^{2}+x+1=0$ has two real roots?

A number a is chosen at random within the interval (-1,1). What is the probability that the quadratic equation $a{x}^{2}+x+1=0$ has two real roots?

For it to have its real roots, we must guarantee that $1-4a\ge 0$, or $a\le \frac{1}{4}$.

A number a is chosen at random within the interval (-1,1). What is the probability that the quadratic equation $a{x}^{2}+x+1=0$ has two real roots?

For it to have its real roots, we must guarantee that $1-4a\ge 0$, or $a\le \frac{1}{4}$.

High school geometryAnswered question

Uroskopieulm 2022-11-14

Derivation of a conditional probability mass function which involves Geometric and Gamma random variables

Let N be a geometric random variable with parameter p. Suppose that the conditional distribution of X given that $N=n$ is the gamma distribution with parameters n and $\lambda .$ Find the conditional probability mass function of N given that $X=x.$.

Let N be a geometric random variable with parameter p. Suppose that the conditional distribution of X given that $N=n$ is the gamma distribution with parameters n and $\lambda .$ Find the conditional probability mass function of N given that $X=x.$.

High school geometryAnswered question

Ty Moore 2022-11-13

Geometric Distribution - Probability

How we can proof that if $X\sim Geo(p)$

Than: $Pr(X>a)=(1-p{)}^{a}$

How can we proof it formally? There is an example for that to understand it intuitively?

How we can proof that if $X\sim Geo(p)$

Than: $Pr(X>a)=(1-p{)}^{a}$

How can we proof it formally? There is an example for that to understand it intuitively?

High school geometryAnswered question

akuzativo617 2022-11-11

Geometric distribution with multiple successes

"A sales representative vows to keep knocking on doors until he makes two sales. Given that his probability of success is u, let X = the number of doors he knocks on.

Find the probability mass function of X"

My thought is that x cannot be less than 2, since he would have to knock on two doors to make two sales.

I'm thinking the function would be ${\textstyle (}\genfrac{}{}{0ex}{}{x}{2}{\textstyle )}({u}^{2})(1-u{)}^{x-2}.$

But when I go to find E(x), that doesn't lend itself well to the geometric form I've learned to love.

Am I on the right track at least?

"A sales representative vows to keep knocking on doors until he makes two sales. Given that his probability of success is u, let X = the number of doors he knocks on.

Find the probability mass function of X"

My thought is that x cannot be less than 2, since he would have to knock on two doors to make two sales.

I'm thinking the function would be ${\textstyle (}\genfrac{}{}{0ex}{}{x}{2}{\textstyle )}({u}^{2})(1-u{)}^{x-2}.$

But when I go to find E(x), that doesn't lend itself well to the geometric form I've learned to love.

Am I on the right track at least?

High school geometryAnswered question

Aden Lambert 2022-11-11

On a certain discrepancy measure between probability distributions on the symmetric group of permutation ${\mathfrak{S}}_{n}$

Let ${\mathfrak{S}}_{n}$ be the symmetric group of permutations on n objects and let P and Q be a probability distributions on ${\mathfrak{S}}_{n}$ (i.e P and Q are points on the n!-simplex). For $1\le i<j\le n$, let ${p}_{ij}$ be the probability that a random permutation $\sigma $ drawn from P ranks j ahead of i, i.e satisfies $\sigma (i)<\sigma (j)$. Consider the quantity $\mathrm{\Delta}(P,Q):=\sum _{1\le i<j\le n}|{p}_{ij}-{q}_{ij}|$.

Is it possible to reasonably upper-bound $\mathrm{\Delta}(P,Q)$ in terms of some distance (e.g total variation) between P and Q ?

Let ${\mathfrak{S}}_{n}$ be the symmetric group of permutations on n objects and let P and Q be a probability distributions on ${\mathfrak{S}}_{n}$ (i.e P and Q are points on the n!-simplex). For $1\le i<j\le n$, let ${p}_{ij}$ be the probability that a random permutation $\sigma $ drawn from P ranks j ahead of i, i.e satisfies $\sigma (i)<\sigma (j)$. Consider the quantity $\mathrm{\Delta}(P,Q):=\sum _{1\le i<j\le n}|{p}_{ij}-{q}_{ij}|$.

Is it possible to reasonably upper-bound $\mathrm{\Delta}(P,Q)$ in terms of some distance (e.g total variation) between P and Q ?

High school geometryAnswered question

nyle2k8431 2022-11-10

Let E be the event of an even number of successes.

${u}_{n}$: Probability of E occurring at the nth trial not necessarily for the first time

${f}_{n}$: Probability of E occurring at the nth trial for the first time

Let U(x) and F(x) be the corresponding probability generating functions and from that we have the equation

$U(x)=1/(1-F(x))$

${u}_{n}$: Probability of E occurring at the nth trial not necessarily for the first time

${f}_{n}$: Probability of E occurring at the nth trial for the first time

Let U(x) and F(x) be the corresponding probability generating functions and from that we have the equation

$U(x)=1/(1-F(x))$

High school geometryAnswered question

undergoe8m 2022-11-09

How come if X is a Geometric random variable, then $P(X>x)={q}^{x}?$

High school geometryAnswered question

inurbandojoa 2022-11-08

An upper bound on the expected value of the square of random variable dominated by a geometric random variable

Let X and Y be two random variables such that:

1. $0\le X\le Y$

2. Y is a geometric random variable with the success probability p (the expected value of Y is 1/p).

I would be grateful for any help of how one could upperbound $\mathbb{E}({X}^{2})$ in terms of p.

Let X and Y be two random variables such that:

1. $0\le X\le Y$

2. Y is a geometric random variable with the success probability p (the expected value of Y is 1/p).

I would be grateful for any help of how one could upperbound $\mathbb{E}({X}^{2})$ in terms of p.

High school geometryAnswered question

inurbandojoa 2022-11-08

Probability that at least 4 out of 365 times I try, I need more than 5 flips before getting a tails

Suppose I flip a coin until I get a tails. Let X be the number of flips this takes. What is the probability that there are at least 4 days in a year where I needed more than 5 flips to get a tails?

Is this kind of question combining geometric and exponential distributions with some kind of conditional probability? I'm unsure how to compute this.

Suppose I flip a coin until I get a tails. Let X be the number of flips this takes. What is the probability that there are at least 4 days in a year where I needed more than 5 flips to get a tails?

Is this kind of question combining geometric and exponential distributions with some kind of conditional probability? I'm unsure how to compute this.

High school geometryAnswered question

Emmanuel Giles 2022-11-08

An identity involving geometric distribution

Let $(\mathrm{\Omega},\mathfrak{A},P)$ be probability space and $X:\mathrm{\Omega}\to {\mathbb{N}}_{\mathbb{0}}$ be a geometrically distributed random variable with parameter $p\in (0,1)$ $X\backsim \text{Geo}(p)$. Show that $P(X=m+n\text{}|X\ge m)=P(X=n),\text{for all}m,n\in {\mathbb{N}}_{\mathbb{0}}.$

Let $(\mathrm{\Omega},\mathfrak{A},P)$ be probability space and $X:\mathrm{\Omega}\to {\mathbb{N}}_{\mathbb{0}}$ be a geometrically distributed random variable with parameter $p\in (0,1)$ $X\backsim \text{Geo}(p)$. Show that $P(X=m+n\text{}|X\ge m)=P(X=n),\text{for all}m,n\in {\mathbb{N}}_{\mathbb{0}}.$

High school geometryAnswered question

Ty Moore 2022-11-07

Why is this not a geometric distribution?

A recruiting firm finds that 20% of the applications are fluent in both English and French. Applicants are selected randomly from a pool and interviewed sequentially. Find the probability that at least five applicants are interviewed before finding the first applicant who is fluent in both English and French

The answer for this question is $(0.8{)}^{5}$.

A recruiting firm finds that 20% of the applications are fluent in both English and French. Applicants are selected randomly from a pool and interviewed sequentially. Find the probability that at least five applicants are interviewed before finding the first applicant who is fluent in both English and French

The answer for this question is $(0.8{)}^{5}$.

High school geometryAnswered question

perlejatyh8 2022-11-05

What is the probability of maximum of two iid geometric random variable?

- Let X,Y be independent geometric random variables, where both are having same parameter (p).

- Let $Z=max(X,Y)$.

I would like to find $P(Z=i)$ for some real values of i.

As we know for $K=min(X,Y)$, K is geometric distributed with parameter $(2p-{p}^{2})$. Does Z also geometric distributed ?.

- Let X,Y be independent geometric random variables, where both are having same parameter (p).

- Let $Z=max(X,Y)$.

I would like to find $P(Z=i)$ for some real values of i.

As we know for $K=min(X,Y)$, K is geometric distributed with parameter $(2p-{p}^{2})$. Does Z also geometric distributed ?.

High school geometryAnswered question

Juan Lowe 2022-11-05

Why can this question be treated as a question with geometric random variable without any modifications?

In an infinite sequence of flipping a fair coin - 0.5 probability to get heads/tails. (every flip is independent from the others).What is the expected value of number of flips until we get Heads then Tails?

In an infinite sequence of flipping a fair coin - 0.5 probability to get heads/tails. (every flip is independent from the others).What is the expected value of number of flips until we get Heads then Tails?

Geometric probability is a branch of mathematics which deals with the probability of certain geometrical shapes forming or intersecting. In geometric probability, a variety of shapes are studied, such as circles, triangles, and lines. These shapes are then used to calculate the probability of certain events taking place. For example, one can calculate the probability of two circles intersecting, or the probability of a line segment hitting a circle. Geometric probability can be used to calculate the probability of many real-world events, such as the probability of a certain object being hit by a thrown ball. With the help of geometric probability, it is possible to calculate the likelihood of various events in the physical world.