# Geometric probability problems with answers

Recent questions in Geometric Probability
ferdysy9 2022-08-16

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

cortejosni 2022-08-16

### X be a geometric random variable, show that $P\left[X\ge n\right]=\left(1-p{\right)}^{n-1}$

lexi13xoxla 2022-08-15

### Are there other similar discrete probability distributions where the probability of random variable Y taking on value n is given by e.g. the n-th term in the series expansion of a function divided by the closed form of the function?Conversely, does this mean that every function with a series expansion has a corresponding discrete probability distribution? Which are the more commonly known functions and their corresponding probability distributions?

Sandra Terrell 2022-08-15

### Conditional Probability and Maximum values of random variables including a Geometric Random VariableLet ${X}_{1},{X}_{2},...$ be i.i.d. random variables with the common CDF F , and suppose they are independent of N, a geometric random variable with parameter p. Let $M=max\left\{{X}_{1},...,{X}_{N}\right\}$(a) Find $Pr\left\{M\le x\right\}$ by conditioning on N.(b) Find $Pr\left\{M\le x|N=1\right\}$.(c) Find $PrM\le x|N>1$.Could anyone explain why in (c)$Pr\left\{M\le x|N>1\right\}=F\left(x\right)Pr\left\{M\le x\right\}$

orkesruim40 2022-08-14

### Average angle between two randomly chosen vectors in a unit squareConsider two randomly chosen vectors (a,b) and (c,d) within the unit square, where a,b,c, and d are chosen uniformly from [0,1]. What is the expected angle between the vectors?

muroscamsey 2022-08-14

### Geometric or binomial distribution?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,...}, $E\left(Z\right)=$ Summation from $i=1$ to infinity of $P\left(Z\ge i\right)$.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 , but I don't know how I will calculate $X<9$, I would know how to calculate $P\left(X=9\right)$, I don't know how to do b or c.

Taliyah Reyes 2022-08-13

### Geometric Probability problem in 3 unknownsSuppose we have to choose 3 numbers, a,b and c such that $a,b,c\in \left[0,1\right]$. The numbers are randomply distributed in an uniform distribution between 0 and 1. Then I've been asked to find the probability of $a+b>2c$.I'm not being able to represent this in a geometrical way. I've tried fixing the value of c and then figuring out where a and b would lie on a line segment, but that got me nowhere.How should I approach this particular type of problem?

Carsen Patel 2022-08-13

### Is there any way to calculate harmonic or geometric mean having probability density function?I have probability density of function of some data (it's triangular.) How can I calculate harmonic or geometric mean of the data? I know for calculating arithmetic mean of a variable like K, I have to calculate ${\int }_{0}^{\mathrm{\infty }}K.P\left(K\right)dK$ but I don't have any ideas for other types of averaging methods (Harmonic and geometric).

heelallev5 2022-08-13

### Geometric or PoissonDrought 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)

Yair Valentine 2022-08-13

### Probability of a triangle inside a squareIf we have the square with vertices at the 4 corners of $\left(0,1{\right)}^{2}$, 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?

Passafaromx 2022-08-12

### Geometric probabilities solution verificationAn 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) 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 ${\left(\frac{1}{2}\right)}^{n}\cdot \left(n-1\right)$. Can someone pleas explain what I did incorrectly and where the $n-1$ factor is coming from?

zabuheljz 2022-08-12

### Finding the probability of getting no successes in a Geometric DistributionIn 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:$P\left(X=30\right)=\left(1-p{\right)}^{30-1}p$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.

ferdysy9 2022-08-12

### Probability of intersection of two geometrical figures in bounded space?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?

Ashlynn Hale 2022-08-12

### 5 independent traffic lights, how many is car expected to pass without getting stoppedThere 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)?

Flambergru 2022-08-12

### MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Find the sum of a probability of dice roll that is prime.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.

motsetjela 2022-08-12

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

dredyue 2022-08-11