Given that a minesweeper has encountered exactly 5 landmines in a particular 10 mile stretch, what is the probability that he will encounter exactly 6 landmines on the next 10 mile stretch. (Average number of landmines is 0.6 per mile in the 50 mile stretch)

Elianna Lawrence 2022-07-14 Answered
Calculating conditional probability given Poisson variable
I encountered a set of problems while studying statistics for research which I have combined to get a broader question. I want to know if this is a solvable problem with enough information specifically under what assumptions or approach.
Given that a minesweeper has encountered exactly 5 landmines in a particular 10 mile stretch, what is the probability that he will encounter exactly 6 landmines on the next 10 mile stretch. (Average number of landmines is 0.6 per mile in the 50 mile stretch)
I have figured that the approach involves finding out the Poisson probabilities of the discrete random variable with the combination of Bayes Conditional probability. But am stuck with proceeding on applying the Bayes rule. i.e Pr ( X = 6 X = 5 ).
I know that Pr ( X = 5 ) = e 6 5 6 / 5 ! . Here λ = 0.6 10 and X = 5) Similarly for Pr ( X = 6 ). Is Bayes rule useful here: P ( Y A ) = Pr ( A Y ) Pr ( Y ) / ( Pr ( A Y ) Pr ( Y ) + Pr ( A N ) Pr ( N ) )?
Would appreciate any hints on proceeding with these types of formulations for broadening my understanding.
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)

Tamoni5e
Answered 2022-07-15 Author has 14 answers
Step 1
The answer to your question depends on what exactly you mean when you say that the "average number of landmines is 0.6 per mile in the 50-mile stretch". This could be taken to mean two different things:
a) You have general knowledge about the density of landmines in this general area, which is 0.6 landmines per mile, but you don't know precisely how many landmines are in this 50-mile stretch.
b) There are exactly 30 landmines in this 50-mile stretch, for an average of 0.6 per mile.
Step 2
If what you mean is a), Each 10-mile stretch can be independently modeled using a Poisson distribution, and since you only know the general density of the landmines but not how many are in this 50-mile stretch, the fact that 5 were encountered in the first 10-mile stretch tells you nothing about how many you'll encountered in the next 10-mile stretch.
If what you mean is b), then the fact that you encountered only 5 and not 6 landmines in the first 10-mile stretch does make a difference. Whereas without this information, you would use a density of 0.6 landmines per mile to model the second 10-mile stretch, and thus the expected value would be 10 0.6, now you know that there are 30 5 = 25 landmines in the remaining 40 miles, so the expected number for the next 10-mile stretch is 25/4 instead of just 24/4, so the conditional distribution given that information is a Poisson distribution with parameter λ = 25 / 4
Not exactly what you’re looking for?
Ask My Question

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 2022-07-21
If I found that a series converges, how can I know to what number it's converging to?
I started learning series in calculus and I have trouble catching a basic concept. When I try to find if a series converges or diverges I have many ways to go about it. If I see that the series diverges than I stop there. If I see that the series converges than there is a number it's converging to right?
For example: 2 n 3 + 4 . I do the limit comparison test with the series 1 n 3 and get a finite number 2. I know that 1 n 3 converges, so now I know that 2 n 3 + 4 converges also. How do I know to what number it converges to?
asked 2022-07-23
A random sample of 5,000 people is selected from local telephone book to participate in financial planning survey to help infer about typical saving and spending habits. three thousand return the questionnaire. Describe the types of bias that may result in at least 4-5 complete sentences.
asked 2022-07-19
Which statement is true with regard to non-response bias?
1.Respondents have strong opinions about the subject matter.
2.Respondents change their answers to influence the results.
3.Respondents give the answer they think the questioner wants.
4.Respondents do not give their honest opinion.
asked 2022-07-17
Find the find the quadratic average and the geometric average. To do this I have these informations :
The standart deviation, the arithmetic average and the number of values.
asked 2022-07-23
From 10 numbers a , b , c , . . . j all sets of 4 numbers are chosen and their averages computed. Will the average of these averages be equal to the average of the 10 numbers?
asked 2022-07-23
In Traditional Survey Reseach, elaborate what are;
1. Nonresponse Bias
2. Refusal rate
3. Response Bias
asked 2022-07-14
Probability of 1 billion monkeys typing a sentence if they type for 10 billion years
Suppose a billion monkeys type on word processors at a rate of 10 symbols per second. Assume that the word processors produce 27 symbols, namely, 26 letters of the English alphabet and a space. These monkeys type for 10 billion years. What is the probability that they can type the first sentence of Lincoln’s “Gettysburg Address”?
Four score and seven years ago our fathers brought forth on this continent a new nation conceived in liberty and dedicated to the proposition that all men are created equal.
Hint: Look up Boole’s inequality to provide an upper bound for the probability!
This is a homework question. I just want some pointers how to move forward from what I have done so far. Below I will explain my research so far.
First I calculated the probability of the monkey 1 typing the sentence (this question helped me do that); let's say that probability is p:
P ( Monkey 1 types our sentence ) = P ( M 1 ) = p
Now let's say that the monkeys are labeled M 1 to M 10 9 , so given the hint in the question I calculated the upper bound for the probabilities of union of all P ( M i ) (the probability that i-th monkey types the sentence) using Boole's inequality.
Since P ( M i ) = P ( M 1 ) = p,
P ( i M i ) i = 1 10 9 P ( M i ) = 10 9 p = 10 9 p
Am I correct till this point? If yes, what can I do more in this question? I tried to study Bonferroni inequality for lower bounds but was unsuccessful to obtain a logical step. If not, how to approach the problem?

New questions