Exercises: Random experiments and sample space

When rolling a $6$-sided die, classify these events as certain, possible or impossible:

1. a) Rolling a $5$.
2. b) Rolling a $7$.
3. c) Rolling a number less than $7$.
4. d) Rolling a number greater than $2$.

Which of these experiments are random? Why?

1. a) Drawing a card from a well-shuffled pack and recording its suit.
2. b) Measuring the length of your notebook with a ruler.
3. c) Recording the number of goals in your team's next match.
4. d) Calculating the perimeter of a table given its dimensions.

Write down the sample space (all possible outcomes) for these experiments:

1. a) Rolling a die and recording the score.
2. b) Flipping a coin and recording the outcome.
3. c) Drawing a ball at random from a tombola drum with numbers from $1$ to $5$.

In a box there are $6$ trading cards: $4$ of players and $2$ of team crests. If we pick a card at random, classify these events as certain, possible or impossible:

1. a) Picking a player card.
2. b) Picking a manager card.
3. c) Picking a player or team crest card.
4. d) Picking a team crest card.

Cian states the following in class about three different situations:

1. "Dropping a fresh egg from a third floor onto the ground is a random experiment, because we cannot predict exactly how many pieces the shell will break into."

2. "In a bag with $5$ green marbles numbered from $1$ to $5$ and $1$ red marble with the number $6$, the sample space is $E = \{\text{green}, \text{red}\}$, so drawing red is just as likely as drawing green."

3. "In that same bag, the event 'drawing a number less than $10$' is a possible but not certain event, because it depends on luck."

1. a) Explain why statement 1 contains a conceptual error: distinguish between the essential outcome of the phenomenon (that the egg will break due to the laws of physics, making it deterministic) and the secondary details.
2. b) Correct statement 2: write the full sample space if the experiment consists of recording the color and the number of the drawn marble, and explain why Cian's reasoning about probability is flawed.
3. c) Correct statement 3: define what a certain event is and justify what type of event "drawing a number less than $10$" actually is in that bag.

For your school's end-of-year party, you are designing a circular wheel of fortune divided into numbered sections. You must choose which numbers to place in the sections while simultaneously fulfilling these four conditions:

1. The sample space $E$ of the numbers that can come up must have exactly $4$ different possible outcomes.

2. The event "getting an odd number" must be a certain event.

3. The event "getting a number greater than $30$" must be an impossible event.

4. The event "getting a multiple of $5$" must be a possible, but not certain, event.

1. a) Propose a sample space $E$ that simultaneously satisfies all four conditions.
2. b) Justify step by step why your proposal fulfills each of the given conditions.
3. c) If you added the number $18$ to your wheel, which of the previous conditions would no longer be met? Explain your reasoning.

State whether each of the following experiments is random or deterministic, justifying your answer:

1. a) Calculating the cost of $3$ kg of oranges at $\text{€}\ 2.10$ per kilo.
2. b) Recording the outcome of dropping a drawing pin and seeing if it lands point up or on its side.
3. c) Measuring the time it takes for a litre of water to boil on the stove.
4. d) Recording who wins Saturday's basketball match.
5. e) Calculating the area of a square with a side length of $5$ cm.

Write down the sample space for each of these random experiments:

1. a) Flipping a coin and recording the outcome.
2. b) Drawing a ball from an urn containing one red, one green, one blue and one yellow ball, and recording its colour.
3. c) Spinning a spinner with numbers from $1$ to $6$ and recording the number.
4. d) Choosing a day of the week at random and recording whether it is a "weekday" or "weekend".

We toss two coins into the air and record whether heads (H) or tails (T) comes up on each one. The figure shows the sample space of the experiment.

1. a) How many possible outcomes does the experiment have? Write them down.
2. b) Are they equally likely? Reason your answer.
3. c) In how many ways can the event "getting at least one head" occur?
4. d) What about the event "getting two of the same"?

A bag contains $9$ red balls and $1$ blue ball. Order the following events for the experiment "picking a ball at random" from most likely to least likely, explaining your criteria:

• Event A: "picking a red ball".

• Event B: "picking a blue ball".

• Event C: "picking a ball that is not green".

• Event D: "picking a green ball".

Met Éireann, the Irish Meteorological Service, issues a snowfall warning for a mountain municipality with three possible alert levels depending on the snow accumulation forecast in $24$ hours: green level (no risk, less than $2\text{ cm}$), yellow level (low risk, between $2$ and $5\text{ cm}$), and orange level (significant risk, more than $5\text{ cm}$).

1. a) Classify the experiment "observing the alert level that is finally activated on Saturday" as random or deterministic. Justify your answer taking into account the nature of weather forecasts.
2. b) Write the sample space $E$ of the alert levels considered for that day.
3. c) Classify the following events as certain, possible, or impossible, justifying each case based on the sample space:

• Event $A$: "The green level or the orange level is activated."

• Event $B$: "The red level (extreme risk) is activated," knowing that for that day Met Éireann has not considered that level.

• Event $C$: "Any of the green, yellow, or orange levels is activated."

1. d) If the authorities decide to suspend school bus routes only if the orange level is activated, is the suspension of school transport a certain, possible, or impossible event on Saturday morning before knowing the final alert? Explain your reasoning.

We roll two tetrahedral dice ($4$-sided dice numbered from $1$ to $4$) and calculate the product of the two numbers obtained.

1. a) Estimate and reason beforehand: without building the full table yet, do you think the sample space of the products will contain more than $10$ different outcomes or fewer than $10$? Do you think obtaining an even product or an odd product will be more frequent? Justify your initial intuition.
2. b) Check your estimate: draw a $4 \times 4$ double-entry table in your notebook where the rows represent the first die and the columns represent the second die, and write in each cell the product of both numbers.
3. c) From the table, write inside braces the sample space $E$ of all the different products that can come up. How many elements does it have? Compare this count with your estimate from part a).
4. d) Count how many cells in the table yield an even product and how many yield an odd product. Was your prior intuition correct? Explain mathematically why in this experiment even outcomes are much more frequent than odd ones.

We roll two dice of different colours (one blue and one green) and record the sum of the scores obtained.

1. a) How many different outcomes can the sum take? Write them all down.
2. b) Are these outcomes equally likely? Check this by completing a two-way table in your notebook showing the $36$ possible pairs (blue die, green die) and counting in how many ways each sum can be obtained.
3. c) Which sum is the most likely? How many ways are there to obtain it?
4. d) And which is the least likely?

State whether the following statements about chance are true or false, giving reasons for your answer:

1. a) Getting a $7$ when rolling a die is an impossible event.
2. b) If an event is very unlikely, then it is impossible.
3. c) If I flip a coin $10$ times, it is certain that heads will come up exactly $5$ times.
4. d) If heads has come up $4$ times in a row, on the next flip tails is more likely "to balance things out".

1. a) Make up a random experiment whose sample space has exactly $6$ equally likely outcomes. Write down its sample space.
2. b) Now make up a random experiment that also has $6$ possible outcomes that are not equally likely, and explain why they are not.
3. c) For your experiment in part b), describe a certain event, an impossible event and an unlikely event.

An opaque bag contains only red and blue balls, $10$ balls in total, but we do not know how many there are of each colour.

1. a) Describe a certain event for this experiment (picking a ball at random).
2. b) Describe an impossible event.
3. c) If you are told that the event "picking a red ball" is very unlikely, what can you deduce about the contents of the bag? Give two possible combinations consistent with this information.
4. d) What if "picking a red ball" were as likely as "picking a blue ball"?

Four schools are participating in a school chess tournament: Rathmines College ($R$), St Brigid's College ($S$), Coláiste Chiaráin ($C$), and Mount Anville College ($M$). For the opening round, a pairing of two schools must be chosen by a random draw (order does not matter: the match $R-S$ is the same as $S-R$). Three students discuss how to determine the sample space of all possible matchups without leaving any out:

• Seán proposes writing combinations directly on the fly as they come to him.

• Aoife proposes building a $4 \times 4$ double-entry table, crossing out the diagonal (a school does not play against itself), and selecting only the cells above the diagonal to avoid duplicating matchups.

• Cian proposes making a two-level tree diagram and then erasing by hand the branches that turn out to be repeated matches.

1. a) Compare the three methods by assessing their order and efficiency: which one of them is the most suitable to guarantee that we do not forget any outcome or count duplicate matches? Justify your choice over the other two.
2. b) Apply the most suitable method you have chosen and write in detail the sample space $E$ of all possible matches. How many elements make up $E$?
3. c) Consider the following events in this experiment and classify them as certain, possible, or impossible, indicating the elements that make up each event:

• Event $A$: "Coláiste Chiaráin participates in the match."

• Event $B$: "Two schools starting with the same letter face each other in the match."

• Event $C$: "At least one of the four registered schools participates in the match."

You are designing a themed deck of cards for a math workshop. You have blank cards and you must write a natural number from $1$ to $20$ on each of them and draw a geometric shape (star or circle). Your goal is to select a set of cards that strictly and simultaneously satisfies the following five conditions:

1. The deck must contain exactly $6$ different cards (this set of $6$ cards will be the sample space $E$ when drawing a card at random).

2. The event "drawing a number that is a multiple of $3$" must be a certain event.

3. The event "drawing a card with a triangle" must be an impossible event.

4. The event "drawing a star with a prime number" must be a possible, but not certain, event.

5. If we define event $A$ as "drawing a number greater than $10$" and event $B$ as "drawing a card with a circle", no element of the sample space can belong to both $A$ and $B$ at the same time (that is, no circle can have a number greater than $10$).

1. a) Design a deck that satisfies all the conditions, detailing explicitly the sample space $E$ with its $6$ elements (indicating the number and shape of each card).
2. b) Prove, analyzing the five conditions one by one, that your proposed sample space satisfies all the requirements.
3. c) Reason mathematically: would it be possible to meet all the conditions if we changed condition 2 to "drawing a number that is both prime and even", keeping the requirement that the deck has $6$ different cards? Justify your answer based on the definition of prime numbers.