Exercises: Random experiments and sample space

State whether each of these experiments is random or deterministic, and justify your answer:

a) Heating water to $100$ °C and checking if it boils.

b) Tossing a coin and recording whether it lands on heads.

c) Measuring how long it takes for a crumpled sheet of paper dropped from a table to hit the ground.

d) Recording the colour of the first car that passes by in the street.

e) Calculating the price of $3$ kg of oranges at $2,40\text{ €}$ per kilo.

In the bag in the figure there are $6$ red balls and $1$ green ball. A ball is drawn at random.

Classify these events as certain, highly likely, unlikely or impossible:

a) Drawing a red ball.

b) Drawing a green ball.

c) Drawing a yellow ball.

d) Drawing a ball that is red or green.

Write down the sample space for each random experiment:

a) Rolling a die and recording the top face.

b) Tossing two coins and recording the number of heads.

c) Drawing a card from a standard Spanish deck of cards and recording the suit.

Write down the sample space for these experiments:

a) Drawing a ball from a drum with balls numbered from $1$ to $10$.

b) Rolling a tetrahedral die with faces numbered from $1$ to $4$.

c) Recording the number of goals a team will score in its next match.

In the experiment 'rolling two dice and adding their scores':

a) Write down a certain event.

b) Write down an impossible event.

c) Write down a possible event that is neither certain nor impossible.

From an urn with balls numbered from $1$ to $8$, a ball is drawn at random. State whether these events are elementary or compound:

a) Drawing $5$.

b) Drawing an even number.

c) Drawing a multiple of $3$.

d) Drawing $8$.

Look at the spinner in the figure.

a) Which colour is most likely to come up when the pointer is spun? And which is least likely?

b) Are there colours with the same probability? Which ones?

c) Order the colours from most likely to least likely.

A coin is tossed and a die is rolled. Copy and complete in your exercise book the sample space table with all possible outcomes:

a) How many different outcomes are there?

b) How many include heads?

c) How many include an even number?

Marta tosses two identical coins in the air and records the total number of heads obtained. When finished, she states out loud: "The sample space of this experiment is $E = \{0, 1, 2\}$; since there are three possible outcomes, each has a probability of one third, so getting two heads is just as likely as getting exactly one head."\n\na) Explain with reasons the mistake Marta makes by confusing the set of possible numerical values with equiprobable elementary events.\nb) Write down the complete sample space detailing the outcomes of each coin (for example, $(H, T)$) and determine how many possible cases actually make up the experiment.\nc) Correct Marta's statement, indicating which of the three outcomes ($0$, $1$, or $2$ heads) is actually the most likely and why.

For a school mathematics fair, you have been commissioned to design a circular wheel of fortune divided into sectors with three colours: red, blue, and white. The random experiment consists of spinning the pointer and recording which colour it stops on.\n\na) Write down the sample space $E$ of the experiment.\nb) Design the distribution of the $360^\circ$ of the circle among the three colours so that these three conditions are met simultaneously:\n 1. The event "get red colour" must be three times as likely to occur as "get white colour".\n 2. The event "get blue colour" must have exactly the same chance of occurring as "get white colour".\n 3. The event "get green colour" must be an impossible event.\nState how many degrees correspond to the sector of each colour.\nc) Write down a certain event for the wheel you have designed and express it in words.

Two dice are rolled and the sum of the scores is recorded.

a) Complete in your exercise book a sample space table with the $36$ possible sums.

b) Write down the sample space of the experiment.

c) How many outcomes give a sum of $7$? List them.

d) Are all outcomes in the sample space equally likely? Explain your answer.

A tetrahedral die (faces numbered $1$ to $4$) and a cubic die (numbered $1$ to $6$) are rolled, and the sum is recorded.

a) Construct a sample space table with all the outcomes.

b) Write down the sample space. How many elements does it have?

c) Write down the outcomes for the event 'getting a sum of $8$'.

You can do it on paper and send a photo with your answer.

Three coins are tossed, as shown in the tree diagram in the figure.

a) How many possible outcomes are there?

b) Write down the sample space.

c) Write down the outcomes for the events 'getting exactly $2$ heads' and 'getting at least one head'.

In the school canteen, you can choose a starter (salad or soup), a main course (chicken, fish or pasta) and a dessert (fruit or yoghurt), as shown in the diagram in the figure.

a) How many different meal combinations can be formed?

b) If a meal is chosen completely at random, how many include pasta?

c) How many include salad and yoghurt?

Three counters are flipped at the same time: one with green and red sides, another with green and blue sides, and another with red and blue sides.

a) Draw a tree diagram and write down the sample space.

b) In how many outcomes are all three colours different?

c) In how many outcomes are there at least two identical colours?

You can do it on paper and send a photo with your answer.

In an urn there are $10$ tokens numbered from $1$ to $10$: those from $1$ to $4$ are red, those from $5$ to $7$ are green and those from $8$ to $10$ are blue. A token is drawn at random. Write down the outcomes for each event:

a) A = 'red token'.

b) B = 'token with an even number'.

c) C = 'blue token with an even number'.

d) D = 'red or green token'.

State whether these statements are true or false, giving reasons:

a) The sample space for tossing two coins and recording the results is $\{CC, C+, ++\}$, and all three outcomes are equally likely.

b) An elementary event consists of a single outcome from the sample space.

c) A certain event has a probability of $1$.

d) When rolling a die, 'getting a number greater than $2$' is a compound event.

In a true-or-false exam with $3$ questions, a student answers completely at random.

a) Using a tree diagram, write down the sample space for their answers, recording for each question whether it is answered correctly ($A$) or incorrectly ($F$).

b) Write down the outcomes for the event 'getting exactly $2$ questions right'.

c) How many outcomes are there for the event 'getting all $3$ questions wrong'?

Two traditional cubic dice (with faces numbered from $1$ to $6$) are rolled at the same time, and the result of the experiment is recorded as the subtraction or positive difference between the two scores: the smaller face is subtracted from the larger one (and if both scores are identical, the result is $0$).\n\na) Estimate: Before making any table or list in your notebook, answer according to your intuition: what do you think is more likely to occur, getting a difference of $0$ or a difference of $1$? Briefly justify what makes you think that.\nb) Construct a $6 \times 6$ double-entry table showing the $36$ resulting differences from the two dice.\nc) Write down the sample space $E$ of the possible differences.\nd) Check: Count in your table how many combinations give a difference of $0$ and how many give a difference of $1$. Compare the actual results with your initial estimate and explain why intuition often fails in this type of comparison.

To study the reliability of forecasts in an agricultural region of inland Spain, the weather of an entire weekend is modelled by only observing whether on Saturday it rains ($R$) or does not rain ($N$), and whether on Sunday it rains ($R$) or does not rain ($N$).\n\na) Reason whether observing what the weather will be like next weekend is a random or deterministic experiment, distinguishing between what the weather forecast predicts and what actually occurs.\nb) Draw a tree diagram representing all possible weather combinations throughout the weekend and write down the sample space $E$ as pairs of outcomes $(S\text{aturday}, S\text{unday})$.\nc) Consider the following three events:\n - Event $A$: "It rains on at least one of the two days".\n - Event $B$: "It does not rain on either of the two days".\n - Event $C$: "It rains three days during the weekend".\nList which outcomes make up each of these events and classify them with justification as possible, certain, or impossible.

Two dice are rolled and their scores are added together. Construct in your exercise book the sample space table with the $36$ possible sums and count the outcomes for each event:

a) Even sum.

b) Sum that is a multiple of $3$.

c) Sum greater than $8$.

d) Sum equal to $1$.

Two dice are rolled and their scores are multiplied together.

a) Construct in your exercise book a sample space table with the $36$ possible products.

b) How many outcomes give an even product?

c) How many outcomes give a product greater than $20$?

d) Which product appears most frequently in the table?

In the experiment in the figure, a coin is tossed: if it lands on heads, a ball is drawn from urn A, and if it lands on tails, from urn B.

a) Write down the sample space for the complete experiment.

b) Are the four outcomes equally likely? Explain your answer.

c) From which urn is it more likely to draw a red ball?

For training, a basketball club has $3$ shirts (red, blue and green), $2$ pairs of shorts (black and white) and $2$ pairs of trainers (model A and model B).

a) Draw a tree diagram and calculate how many different kits can be formed.

b) How many kits include the red shirt?

c) How many include the black shorts and model A trainers?

You can do it on paper and send a photo with your answer.

When rolling two dice and adding their scores, Nora states: 'The sample space is $\{2, 3, 4, \ldots, 12\}$, which is $11$ outcomes, so the probability of getting a sum of $12$ is $\frac{1}{11}$.'

a) Explain the error in her reasoning.

b) Calculate the probability of getting a sum of $12$ correctly.

c) Which sum is the most likely? Why?

The code for a locker at the sports centre has $3$ dials, each with $4$ symbols: circle, triangle, square and star.

a) How many different codes can be formed?

b) How many codes start with a star?

c) How many codes have all three symbols the same?

A set of dominoes has $28$ tiles, each with two numbers from $0$ to $6$, with no pair repeated.

a) How many tiles contain the number $6$? List them.

b) How many tiles are double tiles (both numbers equal)?

c) Describe the event 'the tile contains $6$ or is a double tile' by counting how many outcomes it has.

Compare these two experiments with urns:

• Experiment 1: a ball is drawn at random from a bag containing $3$ green balls and $1$ red ball.

• Experiment 2: a coin is tossed; only if it lands on heads, a ball is drawn from an urn containing $2$ red balls and $1$ green ball.

a) Write down the sample space for each experiment.

b) In which of the two is it easier to obtain a red ball? Reason your answer by calculating and comparing the probability of obtaining a red ball in each one.

In your school's mathematics laboratory, two different random experiments are going to be studied:\n- Experiment 1: A tetrahedral die (with $4$ faces numbered from $1$ to $4$) and an ordinary cubic die (with $6$ faces from $1$ to $6$) are rolled, and the product of the two faces is recorded.\n- Experiment 2: A ball is drawn from an urn containing three balls numbered $1$, $2$, and $3$. If the number drawn is odd, a coin is flipped and it is recorded whether it lands on heads ($H$) or tails ($T$); if, instead, the number drawn is even, a spinner divided into three equal sectors of colours: red ($R$), green ($G$), and blue ($B$), is spun.\n\na) For each of the two experiments, compare and reason which recording method (a double-entry table or a tree diagram) is the most appropriate and clearest to avoid missing any possible cases. Justify your choice by pointing out the difficulties that using the other method would pose in each case.\nb) Apply the method you selected for Experiment 2, represent it in detail, and explicitly write down its sample space $E$. How many elementary events make it up?

You want to build a game for your school's cultural week using two four-sided dice (tetrahedra). Each face must contain a single positive integer (greater than or equal to $1$). The rules of the game require rolling both dice and adding the scores obtained on the resting faces.\nThe mathematics teacher imposes the following strict conditions on you:\n1. The sample space of the sum of both dice must be exactly $E = \{3, 4, 5, 6, 7\}$.\n2. The two dice cannot be identical: the $4$ faces of Die A cannot have exactly the same combination of numbers as the $4$ faces of Die B.\n\na) Deduce step by step which four numbers Die A can have and which four numbers Die B can have (there are several possibilities: giving one is enough) so that the smallest possible outcome is $3$, the largest is $7$, and all intermediate sums can be obtained without any other sum appearing.\nb) Construct the double-entry table with the $16$ possible sum outcomes to verify that your assignment meets all conditions.\nc) From your table, justify whether the $5$ outcomes of the sample space $E$ are equiprobable or not, indicating which sum or sums are repeated the most times.