Random experiments and sample space
Will a coin land on heads? What will two dice add up to? In some games, however much you know, you cannot guess the result before you play. 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 $2,40\text{ €}$ 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:…
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.
Events: certain, possible and impossible
An event is any outcome or set of outcomes of the experiment.
| 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" |
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.
Compound experiments: two dice
When an experiment has several stages (rolling two dice, tossing two coins…), 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:
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?
Back to the start: In a single toss of a coin or roll of a die you cannot know what will come up, because chance is involved, but when you study the sample space of two dice you discover that the most likely total is $7$, with $6$ of the $36$ possible cases.