Hard probability question (cambridge exam question)

In summary, the conversation discusses a question about a desperado in the wild West who is surrounded by an enemy gang and fighting for his survival using several guns. The question asks to show that if the distribution of loaded guns is preserved after each move, then the initial probability of each gun being loaded is p = 3/7. The expected value and variance of the number of loaded guns is also asked to be calculated. Hints are given, including sketching a probability tree for a single gun, and representing the number of loaded guns as a sum of random variables.
  • #1
sweetpotatoes
2
0
original link: http://www.maths.cam.ac.uk/teaching/pastpapers/2001/Part_IA/PaperIA_2.pdf"

Question 11F

Dipkomsky, a desperado in the wild West, is surrounded by an enemy gang and
fighting tooth and nail for his survival. He has m guns, m > 1, pointing in different
directions and tries to use them in succession to give an impression that there are several
defenders. When he turns to a subsequent gun and discovers that the gun is loaded
he fires it with probability 1/2 and moves to the next one. Otherwise, i.e. when the
gun is unloaded, he loads it with probability 3/4 or simply moves to the next gun with
complementary probability 1/4. If he decides to load the gun he then fires it or not with
probability 1/2 and after that moves to the next gun anyway.
Initially, each gun had been loaded independently with probability p. Show that if
after each move this distribution is preserved, then p = 3/7. Calculate the expected value
EN and variance Var N of the number N of loaded guns under this distribution.

Hint: it may be helpful to represent N as a sum Xj (1 to m) of random variables
taking values 0 and 1.

This question is extremely confusing and I don't know even how to start, could anyone help?
 
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  • #2
sweetpotatoes said:
This question is extremely confusing and I don't know even how to start, could anyone help?

To get started on the first part and to help your understanding, try sketching the probability tree for a single gun.

HTH
 
  • #3
bpet said:
To get started on the first part and to help your understanding, try sketching the probability tree for a single gun.

HTH
bpet said:
To get started on the first part and to help your understanding, try sketching the probability tree for a single gun.

HTH

Thanks for the hints.
For m=1, then initially we have prob of p having it loaded, before the first round we have two case
loaded: prob of p
not loaded: 1-p
after 1st round
loaded:
not loaded: 1-p*1/4

Any idea what should I do next?

Also, I don't really understand what he means for "Show that if
after each move this distribution is preserved, then p = 3/7."
What exactly is the "distribution" referring to? is it N?
How could we use the property that the distribution is preserved

Thank you!
 
  • #4
sweetpotatoes said:
... I don't really understand what he means for "Show that if after each move this distribution is preserved, then p = 3/7."

Say the probability after the move is p1, then p1=p. In the first part you'll have derived an expression for p1 in terms of p, and if your algebra is correct then p=3/7 is the only solution to the equation p1=p. To get the probabilities right though, you might need to practice on some simpler probability tree questions first.
 

Related to Hard probability question (cambridge exam question)

1. What is a hard probability question?

A hard probability question is a type of problem that requires advanced mathematical knowledge and critical thinking skills to solve. It often involves complex scenarios and multiple variables, making it challenging to determine the correct answer.

2. How do I approach a hard probability question?

The first step in tackling a hard probability question is to carefully read and understand the given information. Then, you should try to break down the problem into smaller, more manageable parts. It can also be helpful to draw diagrams or make a list of the given information to visualize the problem.

3. What is the difference between probability and statistics?

Probability is the branch of mathematics that deals with the likelihood of events occurring. It involves calculating the chances of an event happening based on the available information. On the other hand, statistics is the study of data and how it can be collected, analyzed, and interpreted to make informed decisions.

4. What are some common strategies for solving hard probability questions?

Some common strategies for solving hard probability questions include using tree diagrams, making tables or charts, and using algebraic equations. It is also essential to understand the fundamental principles of probability, such as the addition and multiplication rules, to apply them correctly in solving the problem.

5. How can I improve my probability skills for exams?

One of the best ways to improve your probability skills is to practice solving different types of probability problems regularly. You can also study and understand the basic principles, definitions, and formulas related to probability. It can also be helpful to work on problems with others and discuss different approaches and strategies for solving them.

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