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Task List 1 — Events and Probability (Sample Spaces)

Visualizing Sample Spaces with Tree Diagrams

Before solving the tasks below, it is often helpful to visualize the experiment using a tree diagram.

A tree diagram represents a random experiment step by step.

  • Each branch represents a possible result of a single step of the experiment.
  • Each level of the tree corresponds to the next stage of the experiment.
  • Each path from the root to a leaf represents one elementary outcome.

This method is especially useful when:

  • the experiment consists of several consecutive steps,
  • the order of outcomes matters,
  • we want to construct the sample space systematically.

In such a diagram:

  • the root represents the start of the experiment,
  • each branching corresponds to the possible outcomes of the next step,
  • each leaf of the tree corresponds to one element of the sample space.

Example — Rock–Paper–Scissors Played Twice

Consider an experiment in which a player chooses one of three options:

  • Rock (\(R\))
  • Paper (\(P\))
  • Scissors (\(S\))

Suppose the choice is made twice in a row, and the order matters.

The experiment can be represented by the following tree diagram.

START
 │
 ├── R
 │     ├── R
 │     ├── P
 │     └── S
 │
 ├── P
 │     ├── R
 │     ├── P
 │     └── S
 │
 └── S
       ├── R
       ├── P
       └── S

Each path from the root to a leaf represents one elementary outcome:

\[ (R,R), (R,P), (R,S), (P,R), (P,P), (P,S), (S,R), (S,P), (S,S) \]

Therefore the sample space contains

\[ 3 \times 3 = 9 \]

elementary outcomes.

Why Tree Diagrams Are Useful

Tree diagrams help to:

  • construct the sample space step by step,
  • clearly see how the number of outcomes grows,
  • distinguish experiments with replacement and without replacement,
  • understand what an elementary outcome represents.

You are encouraged to draw tree diagrams for the experiments in the following tasks whenever possible.

Optional exploration with simulations

Another useful way to understand random experiments is to simulate them on a computer.

You are encouraged to experiment with simple simulations, for example by asking an AI assistant or a chat tool to help you create a small HTML/JavaScript program that performs repeated random trials.

For instance, you could simulate:

  • repeated coin tosses,
  • repeated die rolls,
  • repeated card draws.

Such programs can generate many outcomes and display the results of the experiment.

This type of computational experiment is commonly known as a Monte Carlo simulation.

In a Monte Carlo simulation, the computer performs the same random experiment many thousands or even millions of times.
The results can then be used to estimate probabilities by observing the relative frequencies of events.

When running such simulations, you may notice that:

  • every run of the program produces a different sequence of outcomes,
  • the observed frequencies fluctuate when the number of trials is small,
  • but as the number of trials increases, the frequencies tend to approach the theoretical probabilities that you compute analytically.

For example, when tossing a fair coin the theoretical probability of heads is

\[ P(H) = \frac{1}{2} \]

In a simulation of only a few tosses you might observe:

  • 7 heads out of 10 tosses,
  • or 3 heads out of 10 tosses.

However, if the simulation performs thousands or millions of tosses, the relative frequency of heads will typically become closer and closer to \(0.5\).

It will never match the theoretical value perfectly, but it will usually approach it more closely as the number of trials increases.

When creating your simulations, you may therefore consider adding a feature that allows the program to run very long Monte Carlo experiments, so that you can observe how the empirical frequencies gradually approach the theoretical probabilities calculated in the exercises below.

Task 1 — Coin Tossing

Consider an experiment consisting of tossing a fair coin.

The order of outcomes matters.

  1. Define the sample space \(\Omega_1\) for one coin toss.
  2. Construct the sample space \(\Omega_2\) for two coin tosses.
  3. Construct the sample space \(\Omega_3\) for three coin tosses.
  4. Determine the number of elementary outcomes in each sample space.
  5. Briefly describe what an elementary outcome represents in each case.

Task 2 — Rolling a Die

Consider an experiment consisting of rolling a fair six-sided die.

The order of outcomes matters.

  1. Define the sample space \(\Omega_1\) for one roll of the die.
  2. Construct the sample space \(\Omega_2\) for two consecutive rolls.
  3. Construct the sample space \(\Omega_3\) for three consecutive rolls.
  4. Determine the number of elementary outcomes in each sample space.
  5. Briefly describe what an elementary outcome represents in this experiment.

Task 3 — Drawing Cards

Consider an experiment consisting of drawing cards from a standard 52-card deck.

The order of outcomes matters. Treat each outcome as an ordered sequence of drawn cards.

  1. Define the sample space \(\Omega_1\) for drawing one card.
  2. Construct the sample space \(\Omega_2\) for two consecutive draws with replacement.
  3. Construct the sample space \(\Omega_2'\) for two consecutive draws without replacement.
  4. Determine the number of elementary outcomes in both cases.
  5. Briefly describe what an elementary outcome represents in these experiments.

Task 4 — Weekly Weather Observation

The weather on a given day can be classified into exactly one of the following states:

  • Sunny (\(S\))
  • Cloudy (\(C\))
  • Rainy (\(R\))

The weather is observed once per day for seven consecutive days.

  1. Define the sample space \(\Omega_1\) for the weather observed on one day.
  2. Construct the sample space \(\Omega_2\) for two consecutive days.
  3. Define the sample space \(\Omega_7\) describing the weather observed during seven consecutive days.
  4. Determine the number of elementary outcomes in each sample space.
  5. Briefly describe what an elementary outcome represents in the case of a weekly observation.

Task 5 — Buffon's Needle Experiment

Consider an experiment in which a needle of length \(L\) is thrown randomly onto a floor with equally spaced parallel lines.
The distance between neighboring lines is \(d\).

  1. Describe the sample space \(\Omega\) of this experiment.

  2. Identify the parameters that determine the outcome of a single throw.

  3. Represent an elementary outcome using appropriate variables describing:

  4. the position of the needle relative to the nearest line,

  5. the orientation angle of the needle.

  6. Express the sample space \(\Omega\) as a set of possible values of these variables. (You may restrict to \(x \in [0, \tfrac{d}{2}]\) for the distance of the needle's center from the nearest line and \(\theta \in [0, \tfrac{\pi}{2}]\) for the angle, using symmetry.)

  7. Briefly explain why the sample space in this experiment is continuous, unlike the sample spaces in the previous tasks.

Task 6 — Events and Probabilities in Coin Tossing

Refer to Task 1, where the sample spaces for one, two, and three coin tosses were defined.

Assume the coin is fair, so all elementary outcomes are equally likely.

First assign probabilities to all elementary outcomes in the sample spaces:

  • \(\Omega_1\) (one toss),
  • \(\Omega_2\) (two tosses),
  • \(\Omega_3\) (three tosses).

Then describe the following events as subsets of the sample space and compute their probabilities.

One Coin Toss

Compute the probability of the following events:

  • \(A_1\) — the result is heads,
  • \(B_1\) — the result is tails,
  • \(C_1\) — the result is not tails.

Two Coin Tosses

Compute the probability of the following events:

  • \(A_2\) — exactly one head occurs,
  • \(B_2\) — at least one head occurs,
  • \(C_2\) — both tosses give the same result.

Three Coin Tosses

Compute the probability of the following events:

  • \(A_3\) — exactly two heads occur,
  • \(B_3\) — at least one tail occurs,
  • \(C_3\) — all three tosses give the same result.

Finally, define one additional event on \(\Omega_3\) and compute its probability.

Task 7 — Events and Probabilities in Die Rolling

Refer to Task 2, where the sample spaces for one, two, and three rolls of a fair six-sided die were defined.

Assume the die is fair, so all elementary outcomes are equally likely.

First assign probabilities to all elementary outcomes in the sample spaces:

  • \(\Omega_1\) (one roll),
  • \(\Omega_2\) (two rolls),
  • \(\Omega_3\) (three rolls).

Then describe the following events as subsets of the sample space and compute their probabilities.

One Die Roll

Compute the probability of the following events:

  • \(A_1\) — the result is even,
  • \(B_1\) — the result is greater than 4,
  • \(C_1\) — the result is at most 3.

Two Die Rolls

Compute the probability of the following events:

  • \(A_2\) — the sum of the results equals 7,
  • \(B_2\) — both results are the same,
  • \(C_2\) — the sum of the results is at least 10.

Three Die Rolls

Compute the probability of the following events:

  • \(A_3\) — the sum of the results equals 10,
  • \(B_3\) — exactly two rolls give the same number,
  • \(C_3\) — the outcomes contain two twos and one three (in any order).

Finally, define one additional event on \(\Omega_3\) and compute its probability.

Task 8 — Events and Probabilities in Card Drawing

Refer to Task 3, where the sample spaces for drawing cards from a standard 52-card deck were defined.

Assume the deck is well-shuffled. In each experiment, every ordered sequence of draws is equally likely (with or without replacement as specified).

First assign probabilities to all elementary outcomes in the sample spaces:

  • \(\Omega_1\) (one card drawn),
  • \(\Omega_2\) (two cards drawn with replacement),
  • \(\Omega_2'\) (two cards drawn without replacement).

Then describe the following events as subsets of the sample space and compute their probabilities.

One Card Drawn

Compute the probability of the following events:

  • \(A_1\) — the card is a heart,
  • \(B_1\) — the card is a king,
  • \(C_1\) — the card is not a face card (not J, Q, or K).

Two Cards Drawn (with replacement)

Compute the probability of the following events:

  • \(A_2\) — both cards are hearts,
  • \(B_2\) — both cards have the same rank,
  • \(C_2\) — at least one card is an ace.

Two Cards Drawn (without replacement)

Compute the probability of the following events:

  • \(A_3\) — both cards are hearts,
  • \(B_3\) — both cards have the same rank,
  • \(C_3\) — one card is a king and the other is a queen (in any order).

Finally, define one additional event on \(\Omega_2'\) and compute its probability.

Task 9 — Events and Probabilities in Weekly Weather Observation

Refer to Task 4, where the sample space \(\Omega_7\) describing the weather during seven consecutive days was defined.

Each day can be in exactly one of the following states:

  • Sunny (\(S\))
  • Cloudy (\(C\))
  • Rainy (\(R\))

Model the weather as 7 independent days, where each of the three states occurs with probability \(\frac{1}{3}\).

Describe the following events as subsets of \(\Omega_7\) and compute their probabilities.

Events

Compute the probability of the following events:

  • \(A\) — the entire weekend is sunny (Saturday and Sunday are both \(S\)),

  • \(B\) — Wednesday, Thursday, and Friday are all rainy,

  • \(C\) — at least one day during the week is sunny,

  • \(D\) — no rainy day occurs during the entire week,

  • \(E\) — exactly two days during the week are sunny.

Finally, define one additional event on \(\Omega_7\) and compute its probability.

Task 10 — Events and Probabilities in Buffon's Needle Experiment

Refer to Task 5, where the sample space \(\Omega\) of Buffon's needle experiment was defined.

A needle of length \(L\) is thrown randomly onto a plane with equally spaced parallel lines.
The distance between neighboring lines is \(d\).

Assume \(L \le d\). Let \(X \in [0, \tfrac{d}{2}]\) be the distance from the needle's center to the nearest line and \(\theta \in [0, \tfrac{\pi}{2}]\) the angle between the needle and the lines. Assume \(X\) and \(\theta\) are independent and uniformly distributed on these intervals.

Describe the following events and compute their probabilities.

Events

Compute the probability of the following events:

  • \(A\) — the needle intersects one of the lines,

  • \(B\) — the needle does not intersect any line,

  • \(C\) — the angle between the needle and the lines is smaller than \(\frac{\pi}{6}\),

  • \(D\) — the center of the needle falls at a distance less than \(\frac{d}{4}\) from the nearest line,

  • \(E\) — the needle intersects a line and the angle with the lines is greater than \(\frac{\pi}{4}\).

Finally, define one additional event in this experiment and compute its probability.