Random experiments and sample space

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.

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 (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 $5$.

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"?

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.

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 $2$.

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.

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".

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?