Five friends are to show up at a party from 1:00 to 2:00 and are to stay for 6 minutes each. What is the probability that there exists a point in time such that all of the 5 friends meet?

amacorrit80 2022-07-19 Answered
Geometric Probability in Multiple Dimensions
I understand how Geometric Probability works in 1, 2, and 3 dimensions, but is it possible to do these problems in, say, 5 dimensions? For example,
Five friends are to show up at a party from 1:00 to 2:00 and are to stay for 6 minutes each. What is the probability that there exists a point in time such that all of the 5 friends meet?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Izabelle Frost
Answered 2022-07-20 Author has 13 answers
Step 1
Consider the simpler problem of 2 people. Chart on the x axis the time of stay of first person. Chart on the y axis the time of stay of second person. The cartesian product of these intervals together form a rectangle in R 2
Question: When do the two meet, and what does this mean geometrically? It's easy to see that whenever this rectangle intersects the x = y line, then the two people meet (or have an overlapping time), or else if the rectangle is fully below x = y, or fully above they don't meet.
Extending to higher dimensions: When formulated this way, it should be the case that in R n , when the rectangle formed by the cartesian product of the intervals of stay of different people intersects the line x 1 = x 2 = = x n , (or in parametric form, λ [ 1 , 1 , , 1 ] λ) then, all the people have some overlapping time of stay.
Step 2
Probability of overlap: The probability is now the measure of the union of all the rectangles (or hypercubes in R n ) intersecting with the line ( x 1 = x 2 = ) vs the measure of the entire space under consideration, say under normalization, [ 0 , 1 ] × [ 0 , 1 ] × [ 0 , 1 ]

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-07-22
Geometric Probability Distribution, Expected Values
Let X Geometric ( θ ), and let Y = min ( X , 100 ). Compute (a) E(Y) and (b) E ( Y X ).
I know that the Geometric distribution is ( 1 θ ) k 1 θ and I also know how to calculate expected value but I'm confused about what it means that Y = min ( X , 100 )?
asked 2022-07-17
What is the expected volume of the simplex formed by n + 1 points independently uniformly distributed on S n 1 ?
asked 2022-07-22
Find the probability that a geometric random variable X is an even number
Let α be the probability that a geometric random variable X with parameter p is an even number
a) Find α using the identity α = i = 1 P [ X = 2 i ].
b)Find α by conditioning on wether X = 1 or X > 1
asked 2022-08-16
Unbiased sufficient statistic for 1/p of geometric distribution
Suppose that a random variable X has the geometric distribution with unknown parameter p, where the geometric probability mass function is:
f ( x ; p ) = p ( 1 p ) x , x = 0 , 1 , 2 , ; 0 < p < 1.
Find a sufficient statistic T(X) that will be an unbiased estimator of 1/p.
Now I know the population mean is 1 p p and the sufficient statistic for p is a function of i = 1 n X i . But I am unsure on how to proceed any help greatly appreciated!
asked 2022-07-21
Negative Binomial and Geometric Distributions
An actuary has determined that the number of claims per month can take any number 0, 1, 2, 3,... and follows a negative binomial distribution with mean 3 and variance 12. Calculate the probability that the number of claims is at least three but less than six.
So by using some properties of the negative binomial we can derive that p = 0.25, and r = 1. It was my understanding that a geometric distribution is just a negative binomial distribution with r = 1. Given this I tried using the pmf for geometric distribution and got the wrong answer. Can someone explain to me what is going on here.
asked 2022-08-31
Probability that there is an edge between two nodes in a random geometric graph
I have a graph with vertices being generated uniformly over [ 0 , 1 ] 2 . There is an edge between two vertices if the Euclidean distance between the two vertices is r. I am trying to find the probability of this. For that I am starting as below:
P ( two random nodes have an edge between them ) = P ( ( x i x j ) 2 + ( y i y j ) 2 r 2 )
asked 2022-08-18
A mp3 player contains 10 different songs in its memory and chooses songs randomly (every choice is independent on the previous choices).
Let X be the number of songs one will hear until the first time (including this time) the same song will be chosen twice in a row .
A. Write the probability function of X
B. What is the expectation of X?

New questions