Recent questions in Geometric Probability

ferdysy9
2022-08-16

If X follows a geometric distribution, where $p=$ probability of success and $q=$ probability of failure, why is $P(X>r)={q}^{r}?$

cortejosni
2022-08-16

If we choose 2 numbers from [0,2], independent from each other,what is the probability of their geometric mean being higher than 1/6?

Roderick Bradley
2022-08-16

Geometric Probability
Answered

muroscamsey
2022-08-14

A monkey is sitting at a simplified keyboard that only includes the keys "a", "b", and "c". The monkey presses the keys at random. Let X be the number of keys pressed until the money has passed all the different keys at least once. For example, if the monkey typed "accaacbcaaac.." then X would equal 7 whereas if the money typed "cbaccaabbcab.." then X would equal 3.

a.) What is the probability $X\ge 10$?

b.) Prove that for an random variable Z taking values in the range {1,2,3,...}, $E(Z)=$ Summation from $i=1$ to infinity of $P(Z\ge i)$.

c.) What's the expected value of X?

First, is this a binomial distribution or a geometric distribution? I believe it is a binomial but my other friends says that it is geometric. As for the questions above, for a can I just do $1-P(X=9)\text{}\text{or}\text{}1-P(X9)$, but I don't know how I will calculate $X<9$, I would know how to calculate $P(X=9)$, I don't know how to do b or c.

Geometric Probability
Answered

heelallev5
2022-08-13

Drought length is referred to as the number of consecutive time intervals in which the water supply remains below a critical value. Consider the drought length as a random variable, denoted as Y, which is assumed to have a geometric distribution with $p=0.409$

1. What is the probability that a drought lasts exactly 3 intervals? (0.0844)

2. What is the probability that a drought lasts at most 3 intervals? (0.878)

Geometric Probability
Answered

Yair Valentine
2022-08-13

If we have the square with vertices at the 4 corners of $(0,1{)}^{2}$, and we choose a random point z inside the square, the triangle is between (0,0), (1,0) and z, what is the CDF and PDF of the random variable ${A}_{T}$ representing the area of the triangle?

Geometric Probability
Answered

Passafaromx
2022-08-12

An unbiased coin is tossed until a head appears and then tossed until a tail appears. If the tosses are independent, what is the probability that a total of exactly n tosses will be required?

My attempt:

P(n tosses required to produce one head and one tail) $=P(\text{x tosses needed for first head})\times P(\text{y tosses needed for first tail})\text{where}\text{}x+y=n.$

So, the probability becomes $\begin{array}{r}{\left(\frac{1}{2}\right)}^{x-1}\cdot \left(\frac{1}{2}\right)\cdot {\left(\frac{1}{2}\right)}^{y-1}\cdot \left(\frac{1}{2}\right)={\left(\frac{1}{2}\right)}^{n}\end{array}$.

This is not the correct answer, however. The correct answer is ${\left(\frac{1}{2}\right)}^{n}\cdot (n-1)$. Can someone pleas explain what I did incorrectly and where the $n-1$ factor is coming from?

Geometric Probability
Answered

zabuheljz
2022-08-12

In Geometric Distribution, I am getting the probability for doing x number of trials and get my first success with each trial of probability p.

So suppose I want to find what's the probability of doing 30 trials and get my first success on the 30th trial, I do this:

$P(X=30)=(1-p{)}^{30-1}p$

Now, then if I want to find the probability for not getting a single success at all even after doing 30 trials on this same distribution, what should I do? The parameters of the Geometric Distribution doesn't seem to let me find this.

I thought of using like 1 minus the CDF of 30 trials of the geometric distribution but I am not sure if it would be accurate.

Geometric Probability
Answered

ferdysy9
2022-08-12

I'd like to find a closed form (if possible) expression of the probability of interesection of two geometrical figures ${F}_{1}$ and ${F}_{2}$ of area ${A}_{1}$ and ${A}_{2}$, respectively, that are have a random position and orientation in a bounded 2-dimensional space of area ${A}_{tot}$.

Obviously, this probability depends on the exact geometry of ${F}_{1}$, ${F}_{2}$, and the space in which they live. However, is there a closed form expression of this probability for some classes of geometries or shall I go for Monte-Carlo methods?

Geometric Probability
Answered

Ashlynn Hale
2022-08-12

There are 5 independent traffic lights, each with chance of stopping a car equal to 0.6. How many traffic lights is the car expected to pass before being stopped - what is E(X)?

Geometric Probability
Answered

Flambergru
2022-08-12

Consider rolling n fair dice. Let p(n) be the probability that the product of the faces is prime when you roll n dice. For example, when $$ = $\frac{1}{2}$

When $$

When $$ $\frac{1}{24}$. Find the infinite sum of p(n) (when n starts at 0) (Hint: Consider differentiating both sides of the infinite geometric series: infinite sum of $$ when $$)

I can differentiate the two sides of the geometric series but I'm lost regarding what to do after that. I don't fully understand the question.

Geometric Probability
Answered

motsetjela
2022-08-12

X,Y are i.i.d. random variables with mean $\mu $, and taking values in {0,1,2,...}.Suppose for all $m\ge 0$, $P(X=k|X+Y=m)=\frac{1}{m+1}$, $k=0,1,...m$. Find the distribution of X in terms of $\mu $.

Geometric Probability
Answered

dredyue
2022-08-11

If no-one obtains "head", the game continues with the same probabilities as before." If that is the case, why does that affect the probability recursively? Could someone explain why we add the case where no one wins to the probability, and why we multiply it by p?