Problem set 3: Bayes’ Theorem
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
There are three cards. The letter A is written on both sides of the 1st card; the letter A is written on both sides of the 2nd card; letters A and B are written on different sides of the 3rd card.
A random card has been put on the table, such that the letter A can be seen.
What is the probability that the letter A is written on the other side of the card?
A system consists of two parallel elements and is working if at least one of them is working.
At a random time the 1st element is out of order with probability \(0.1\), the 2nd is out of order with probability \(0.2\).
Someone told us that now the system is working. What is the probability that the 2nd element is out of order?
Rachel is going to a party. There is a \(60\%\) chance Ross will go too. If Ross does not go, there is a \(20\%\) chance she will enjoy herself. If Ross does go, there is a \(70\%\) chance she will enjoy herself.
What is the probability that Rachel will enjoy the party?
Suppose you know Rachel did not enjoy herself. What is the probability that Ross was not present?
A student can enter a course either as a beginner (\(73\%\) of all students) or as a transferring student (\(27\%\) of all students). It is found that \(62\%\) of beginners eventually graduate, and that \(78\%\) of transferring students eventually graduate. Find:
the probability that a randomly chosen student is a beginner who will eventually graduate;
the probability that a randomly chosen student will eventually graduate;
the probability that a randomly chosen student is either a beginner or will eventually graduate, or both;
Are the events “Eventually graduates” and “Enters as a transferring student” statistically independent?
If a student eventually graduates, what is the probability that the student entered as a transferring student?
If two entering students are chosen at random, what is the probability that not only do they enter in the same way but that they also both graduate or both fail?
Two snipers shoot a target. Sniper A hits with probability \(0.5\) and sniper B hits with probability \(0.8\). They toss a fair coin to determine who shoots first. Find the probability that sniper A shot first if it is known that after the first shot the target was hit.
A player picks a spot at random within a region \(S\) on a flat surface. \(S\) is split into four sections, each covering \(50\%\), \(30\%\), \(12\%\), and \(8\%\) of the total \(S\) area. If the chosen spot falls into one of these sections, the player wins a prize with probabilities of \(0.01\), \(0.05\), \(0.20\), and \(0.50\), respectively.
The player has now selected a spot and won a prize. Which section of the \(S\) area is the most likely location for the chosen spot?
2 Category 2
The Department of Public Health is giving a free medical test for a certain disease. The test is \(90\%\) reliable in the following sense: if a person has the disease, there is a probability of \(0.9\) that the test will give a positive response; whereas, if a person does not have the disease, there is a probability of only \(0.1\) that the test will give a positive response. Data indicate that your chances of having the disease are only \(1\) in \(10{,}000\). You decide to take the test. A few days later you learn that you had a positive response to the test.
What is the probability that you have the disease?
Consider a high-risk population where \(5\%\) of people have COVID-19. A diagnostic test is correct in \(95\%\) of cases if a person has COVID-19 and in \(90\%\) of cases if a person does not have COVID-19. If a person tests positive, indicating COVID-19, what is the probability that the person does not have COVID-19?
Three dice are rolled. Find the probability of getting \(6\) on at least one die if:
The numbers shown on the three dice are different.
The number shown on the first and on the third die is the same.
In box A there are \(10\) white balls and \(20\) black balls. In box B there are \(10\) white balls and \(10\) black balls. We randomly pick \(4\) balls out of box A and \(6\) balls out of box B and put them all together in an empty box C. What is the probability of picking a white ball from box C?
The exam in probability theory consists of \(n\) questions. \(k\) of these questions are “easy”. Find the probability that in a group of \(n\) students:
The first student chooses an easy question.
The \(i\)-th student chooses an easy question.
There are \(10\) phones in a store, for which the probability of proper functioning within a month is \(0.9\), and \(5\) phones with a similar probability of \(0.95\). Find the probability that two phones, bought randomly from the store, will work properly for a month.
In a box there are \(N\) balls. \(M\) of them are white. We take out \(n\) balls and put them back. Consider the following events:
\[ A_k=\{\text{in the }k\text{-th trial we took a white ball}\}, \]
\[ B_m=\{\text{we took out }m\text{ white balls in total}\}. \]
Find
\[ \mathsf P(A_k\mid B_m). \]
3 Category 3
All tickets (\(100\) tickets) for the Moscow–Berlin flight have been purchased. All passengers arrived for boarding. The first person to enter the plane is an old woman who sits in a random seat. Passengers enter in turn, and each next passenger takes his seat, if it is free. If the seat is occupied, then the passenger sits in a random seat from the remaining ones. What is the probability that the last passenger will take their seat?
Monty Hall problem. Suppose you are on a game show, and you are given the choice of three doors: behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what is behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
Two players conduct an endless series of independent tests. In each test, player \(A\) rolls \(3\) dice and player \(B\) rolls \(2\) dice at the same time. They carry out these tests until they obtain “six” on at least one of the dice. Find the probability of the following event:
- The first “six” was rolled by player A, and not by B.
From an urn containing \(a\) white and \(b\) black balls, two players take turns drawing balls. The winner is whoever chooses the white ball first. Find the probability that the first player wins in cases where the balls are drawn:
under the scheme of equiprobable choice with return;
under the scheme of equiprobable choice without replacement.