Random experiments and sample space
Before a hurling match at Croke Park, the referee tosses a coin to decide who picks the side. Will it land on heads? However much you know, you cannot guess the result before it happens. That has a name in maths: chance.
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…
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 $\text{€}\ 2.40$ per kilo.
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…
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…
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.
Events: certain, possible and impossible
An event is any outcome or set of outcomes of the experiment.
Probability is measured on a scale from $0$ to $1$: a certain event has probability $1$, an impossible event has probability $0$, and a possible event that is neither certain nor impossible has a probability strictly between $0$ and $1$.
To compare how likely two events are, count the outcomes. When all outcomes are equally likely, $P(\text{event}) = \dfrac{\text{favourable outcomes}}{\text{total outcomes}}$. The event with the bigger fraction is more likely. On a spinner, a bigger sector means a more likely colour; with two urns, compare the fractions for each urn.
| Event | What it means | Example with a die |
|---|---|---|
| Certain | Always happens | "getting a number less than 7" |
| Possible | Can happen, but is not certain | "getting a 4" |
| Impossible | Can never happen | "getting an 8" |
Elementary and compound events
An elementary event is a single outcome of the sample space. A compound event is a set of two or more outcomes.
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:…
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.
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…
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.
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…
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.
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…
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 $8$.
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…
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…
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.
Compound experiments: two dice
When an experiment has several stages (rolling two dice, tossing a coin and a die…), a two-way table is used so that no outcome is missed: one row for each result of the first die, one column for each result of the second.
Roll the two dice many times and observe which totals come up most often:
A coin is tossed and a die is rolled. Copy and complete in your exercise book the sample space table with all possible outcomes: | | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |…
A coin is tossed and a die is rolled. Copy and complete in your exercise book the sample space table with all possible outcomes:
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | |
|---|---|---|---|---|---|---|
| C | C$1$ | |||||
| + | +$6$ |
a) How many different outcomes are there?
b) How many include heads?
c) How many include an even number?
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…
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.
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…
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 $1$.
Tree diagrams
When the experiment combines coins or choices with only two or three options at each stage, the tree diagram is clearer than a table: each branch is a possibility, and the final outcomes are the complete paths from the root to the leaves.
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…
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 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…
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?
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;…
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.
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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.
Fundamental principle of counting
You do not always need to list every outcome. If one stage has $m$ options and the next has $n$ options, there are $m \times n$ outcomes altogether. This is the fundamental principle of counting. It works because each option of the first stage branches into $n$ options of the second.
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…
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?
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…
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.
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…
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?
Back to the start: you cannot know how the referee's coin will land, because chance is involved. But with the sample space you know every possible outcome, and with two dice you discover that the most likely total is $7$, with $6$ of the $36$ possible cases.