Problem set 1: Basics of Probability. Combinatorics

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

Suppose that a class of 100 students consists of four subgroups…

Men Women
Taking Economics 17% 38%
Not taking Economics 23% 22%

What is the chance that a randomly chosen student is:

  1. a woman?
  2. taking economics?
  3. a man or taking economics?
  4. a woman and taking economics?

Suppose that in families with three children births are independent, and the probability of a boy on each birth is \(52\%\). Use the table

Outcome BBB BBG BGB BGG GBB GBG GGB GGG
Probability 0.14 0.13 0.13 0.12 0.13 0.12 0.12 0.11

and find the chance that in a family of three children, there will be:

  1. exactly 2 girls;
  2. at least two girls;
  3. at least one child of each sex;
  4. the middle child being opposite in sex to the other two.

You take 3 cards from a deck of 36 cards. After you take each card you return it to the deck and shuffle the deck.

  1. What is the set of elementary outcomes and what are their probabilities?
  2. What is the probability to get \((\text{queen}, \text{king}, \text{ace})\)? The first card is a queen, the second card is a king, the third card is an ace.
  3. What is the probability to get \((\text{king}, \text{king}, \text{ace})\)?

You take 3 cards from a deck of 36 cards. After you take each card you do not return it to the deck.

Answer parts 1, 2, and 3 from the previous problem.

Find unions and intersections of the following events. In which case is one event a subset of the other?

  1. \(A = \{1, 2, 5, 6\}\), \(B = \{1, 5\}\).
  2. \(A = \{\text{Ann}, \text{Mary}, \text{Mike}\}\), \(B = \{\text{Tom}, \text{Mike}, \text{John}\}\).
  3. \(A = \{\text{Moscow}, \text{London}, \text{Paris}\}\), \(B = \{\text{Paris}, \text{Berlin}, \text{Tokyo}\}\), \(C = \{\text{Tokyo}, \text{Rome}\}\).
  1. If \(A\) and \(B\) are mutually exclusive events with probabilities of \(0.6\) and \(0.2\) respectively, then what is the probability of \(A\) or \(B\) occurring?
  2. If \(\mathbb P(A)=0.2\), \(\mathbb P(B)=0.3\), and \(\mathbb P(A\cap B)=0.1\), then what is \(\mathbb P(A\cup B)\)?
  3. If \(\mathbb P(A)=0.4\), \(\mathbb P(B)=0.5\), and \(\mathbb P(A\cup B)=0.7\), then what is \(\mathbb P(\overline{A\cap B})\)?

A survey of the houses in an old residential area found \(30\%\) with holes in the roof, \(40\%\) with broken windows, and \(25\%\) with both problems.

  1. What is the proportion of houses with one or the other, or both, problems?
  2. What is the proportion of houses with holes in the roof but without broken windows?
  3. What is the proportion of houses with exactly one of these problems?
  4. What is the proportion of houses with none of these problems?

Suppose a word is picked at random from this sentence.

  1. What is the sample space of this random experiment?
  2. Find the probability that:
    1. the word has at least 4 letters;
    2. the word contains at least 2 vowels;
    3. the word contains at least 4 letters and at least 2 vowels.

In a group of students, none of whom are from DSBA, \(25\%\) smoke hookah, \(60\%\) drink alcohol, and \(15\%\) do both.

What fraction of students have at least one of these bad habits?

20 families live in a neighborhood of Wolfenstein Castle: 4 have 1 child, 8 have 2 children, 5 have 3 children, and 3 have 4 children.

If we meet a local child near the castle, what are the probabilities \(p_1,p_2,p_3,p_4\) that the child comes from a family with 1, 2, 3, 4 children?

Suppose we roll a red die and a green die.

What is the probability that the number on the red die is larger than the number on the green die?

Two dice are rolled. Find the probability that:

  1. the two numbers will differ by 1 or less;
  2. the maximum of the two numbers will be 5 or larger.

In Galileo’s time people thought that when three dice were rolled, a sum of 9 points and a sum of 10 points had the same probability, since each could be obtained in 6 ways:

\[ \begin{aligned} 9: \quad &1+2+6,\;1+3+5,\;1+4+4,\;2+2+5,\;2+3+4,\;3+3+3,\\ 10: \quad &1+3+6,\;1+4+5,\;2+4+4,\;2+3+5,\;2+2+6,\;3+3+4. \end{aligned} \]

  1. Compute the probability of the event \(1+2+6\): one point on one die, two points on another, and six points on the remaining die.
  2. Compute the probability of the event \(2+4+4\): two points on one die, and four points on each of the remaining dice.
  3. Find the probabilities of the events \(A=\) “a total of 9 points on three dice” and \(B=\) “a total of 10 points on three dice”.

2 Category 2

Suppose \(n\) friends are sitting around a table in random order.

What is the probability that \(A\) sits next to \(B\)?

In a box there are 10 white balls and 6 black balls. 4 balls are chosen randomly without replacement.

What is the probability that among the chosen balls:

  1. there is at least one black ball;
  2. there are exactly two black balls?

A closet contains \(n\) pairs of boots. \(2r\) boots are randomly selected from the closet.

What is the probability that among the selected boots:

  1. there are no complete pairs;
  2. there is exactly one complete pair?

Find the probability that among 50 students attending a lecture on probability theory, at least two of them have the same date of birth.

In a box there are 28 black balls and 4 white balls. 10 balls are chosen randomly.

What is the probability that among the chosen balls:

  1. there is at least one white ball;
  2. there is exactly one white ball;
  3. there are at least two white balls;
  4. there are exactly two white balls?

A deck of playing cards contains 52 cards, divided into 4 different suits of 13 cards each. 6 cards are randomly drawn.

Find the probability that:

  1. among these cards there will be the king of spades;
  2. among these cards there will be a representative of each suit.

3 Category 3

Two \(M\)-sided dice are thrown.

Find the probability that the sum of the two numbers obtained is equal to \(i\).

A set of \(n\) balls is randomly placed into \(m\) boxes.

Find the probability that all boxes are non-empty if the balls are distinguishable.

A shelf holds 12 books in a row. We pick 5 books randomly.

Find the probability that no pair of adjacent books is chosen.

Some residents of Dolgoprudny consider a tram ticket “special” if the sum of the first three digits of its six-digit number is equal to the sum of the last three digits.

Find the probability of getting a “lucky” ticket.