Problem set 2: Probabilistic Models

During the seminars, we will work on problems of three different difficulty levels. Category 1 (green) contains introductory problems, Category 2 (blue) contains standard problems, and Category 3 (orange) contains the most challenging problems.

1 Category 1

Two points \(X\) and \(Y\) are randomly chosen on an interval \(OA=[0,1]\). Find the probability of each of the following events:

  1. A distance between \(X\) and \(O\) is less than \(\dfrac{1}{10}\).
  2. A distance between \(X\) and \(O\) is between \(0.7\) and \(0.705\).
  3. A distance between \(X\) and \(O\) is equal to \(0.7\).
  4. A distance between \(X\) and \(Y\) is less than \(0.5\).
  5. A distance between \(X\) and \(Y\) is equal to \(\dfrac13\).
  6. Length of \(XY\) is less than the distance between \(O\) and the closest point to it.

Consider a round shooting mark of radius \(R\). Someone shoots to it with bullets of radius \(B\). Find the probability that a hole made in the shot entirely lies in an interior circle of radius \(r\). Assume \(R>r>B\).

You have a biased coin for which \(P(H)=p\). You toss the coin \(20\) times. What is the probability that:

  1. you observe first \(8\) heads and then \(12\) tails;
  2. you observe \(8\) heads and \(12\) tails;
  3. you observe more than \(8\) heads and more than \(8\) tails?

Suppose a good password must consist of two lowercase letters (a to z), followed by one capital letter (A to Z), followed by four digits. For example, “ejT3018” is a good password.

  1. Find the total number of good passwords.
  2. A hacker wrote a program that randomly generated \(10^8\) good passwords, where one password could be generated more than once. What is the probability that at least one of the generated passwords matches the password of a particular user?
  3. Answer question 2 assuming that the program generated \(10^8\) distinct passwords.

The student has learned \(20\) out of \(25\) exam questions before the exam. She will be asked \(3\) different questions. If she answers all the questions, she will receive an excellent mark; if she answers \(2\) questions, she will receive a good mark; and if she answers \(0\) or \(1\) question, she will receive an unsatisfactory mark. Find the probability of the following:

  1. obtaining an excellent mark;
  2. receiving an unsatisfactory mark;
  3. passing the exam;
  4. passing the exam if she knows the answer to the first question.

From a brood of mice, containing two white specimens, four mice are taken at random without return. The probability that both white mice were taken is twice as likely as the probability that neither was taken. How many mice are there in the brood?

2 Category 2

Find the probability that in \(2n\) trials of the Bernoulli scheme with probability of success \(p\) and probability of failure \(q=1-p\), there will be \(m+n\) successes and all trials with even numbers will end in success.

A rod of length \(l\) is broken at two randomly chosen points. Find the probability that the obtained segments can form a triangle.

Consider two coins. Let

\[ A=\{\text{the first coin lands heads}\},\qquad B=\{\text{the second coin lands heads}\}, \]

and

\[ C=\{\text{heads appeared only once}\}. \]

Are \(A\), \(B\) and \(C\) mutually independent?

Three dice are thrown. Events \(A\), \(B\) and \(C\) stand for rolling matching numbers, for example two \(6\)’s, on the first and second dice, on the second and third dice, and on the first and third dice, respectively. Are these events pairwise independent and mutually independent?

We randomly order the numbers from \(0\) to \(99\), obtaining the sequence \((x_1,x_2,\ldots,x_{100})\). Are the following events independent: \(\{x_{80}>x_{81}\}\) and \(\{x_{81}>x_{82}\}\)?

A particle changes its position by one unit every second. It moves to the right with probability \(p\) and to the left with probability \(1-p\). Let \(x_t\) correspond to the position of the particle at time \(t\). Find the probability that at time \(t=n\) the particle moved \(k\) units to the right. Let \(x_0=0\).

3 Category 3

A point \(A\) is randomly picked from the inside of a rectangle of sides \(1\) and \(2\). Find the probability of the following events:

  1. The distance from \(A\) to the nearest diagonal is at most \(x\).
  2. The distance from \(A\) to each side is at most \(x\).

Find the probability that three randomly chosen segments of length no more than \(1\) can form a triangle.

Bertrand paradox. Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?

Consider three different methods: the random endpoints method, the random radial point method, and the random midpoint method.

In the Bernoulli scheme, the probability of success, that is, getting \(1\), is \(p<1\). Find the probability that in an infinite sequence of trials:

  1. \(00\) will appear before \(01\);
  2. \(00\) will appear before \(10\).

There are \(a\) white and \(b\) black balls in a box. Someone takes balls from the box, one at a time. Let \(A_k\) be the event that, at time \(k\), he picks a white ball. Are the events \(A_1,\ldots,A_n\) independent?

Consider the Bernoulli scheme. Let \(A_i\) be a random subset of \(\{1,2,\ldots,n\}\). Consider \(m\) such sets. Find:

  1. \(P(A_i\cap A_j=\varnothing)\) for arbitrary \(i,j\);
  2. \(P(A_1\subset A_2\cap A_3)\).

Consider the Bernoulli scheme. Let \(A_i\) be a random subset of \(\{1,2,\ldots,n\}\). Consider \(m\) such sets. Find \(P(|A_1\cup\cdots\cup A_m|=k)\).