Random experiments and sample space
At recess, Diego and Ximena decide who gets the last sandwich with a coin flip: heads or tails. No matter how much math they know, they can't guess what will come up before the coin is tossed. That has a name: chance. On this page you'll learn how to describe what can happen in an organized way and how to compare how likely each outcome is.
Random and deterministic experiments
Which of these experiments are random? Why? a) Drawing a card from a well-shuffled pack and recording its suit. b) Measuring the length of your notebook with a ruler. c)…
Which of these experiments are random? Why?
- a) Drawing a card from a well-shuffled pack and recording its suit.
- b) Measuring the length of your notebook with a ruler.
- c) Recording the number of goals in your team's next match.
- d) Calculating the perimeter of a table given its dimensions.
State whether each of the following experiments is random or deterministic, justifying your answer: a) Calculating the cost of $3$ kg of oranges at $\$42…
State whether each of the following experiments is random or deterministic, justifying your answer:
- a) Calculating the cost of $3$ kg of oranges at $\$42$ per kilo.
- b) Recording the outcome of dropping a thumbtack and seeing if it lands point up or on its side.
- c) Measuring the time it takes for a liter of water to boil on the stove.
- d) Recording who wins Saturday's basketball game.
- e) Calculating the area of a square with a side length of $5$ cm.
Sample space
Click "Roll" many times and check that the die always lands on one of those 6 outcomes, never any other:
Write down the sample space (all possible outcomes) for these experiments: a) Rolling a die and recording the score. b) Flipping a coin and recording the outcome. c)…
Write down the sample space (all possible outcomes) for these experiments:
- a) Rolling a die and recording the score.
- b) Flipping a coin and recording the outcome.
- 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: a) Flipping a coin and recording the outcome. b) Drawing a ball from an urn containing one red, one…
Write down the sample space for each of these random experiments:
- a) Flipping a coin and recording the outcome.
- b) Drawing a ball from an urn containing one red, one green, one blue and one yellow ball, and recording its color.
- c) Spinning a spinner with numbers from $1$ to $6$ and recording the number.
- d) Choosing a day of the week at random and recording whether it is a "weekday" or "weekend".
Two-step experiments: tables and trees
If you flip two coins, the result is a pair: what came up on the first and what came up on the second. To avoid forgetting any outcome, there are two tools.
Tree diagram. From each outcome of the first coin, branches lead to the outcomes of the second.
Two-way table. Rows for the first coin, columns for the second:
| 1st \ 2nd | H | T |
|---|---|---|
| H | HH | HT |
| T | TH | TT |
There are $4$ outcomes: $E = \{HH, HT, TH, TT\}$. Notice that $HT$ and $TH$ are different: it matters which coin gave which result.
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. a) How many possible…
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.
- a) How many possible outcomes does the experiment have? Write them down.
- b) Are they equally likely? Explain your answer.
- c) In how many ways can the event "getting at least one head" occur?
- d) What about the event "getting two of the same"?
We roll two dice of different colors (one blue and one green) and record the sum of the scores obtained. a) How many different outcomes can the sum take? Write them all…
We roll two dice of different colors (one blue and one green) and record the sum of the scores obtained.
- a) How many different outcomes can the sum take? Write them all down.
- 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.
- c) Which sum is the most likely? How many ways are there to obtain it?
- d) And which is the least likely?
Events: certain, possible, and impossible
An event is an outcome or a set of outcomes of the experiment (for example, "rolling an even number").
| Event | What it means | Example with a die |
|---|---|---|
| Certain | Always happens | "rolling a number less than 7" |
| Possible | Can happen, but isn't certain | "rolling a 4" |
| Impossible | Can never happen | "rolling an 8" |
When rolling a $6$-sided die, classify these events as certain, possible or impossible: a) Rolling a $5$. b) Rolling a $7$. c) Rolling a number less than $7$. d) Rolling…
When rolling a $6$-sided die, classify these events as certain, possible or impossible:
- a) Rolling a $5$.
- b) Rolling a $7$.
- c) Rolling a number less than $7$.
- 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: a)…
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:
- a) Picking a player card.
- b) Picking a manager card.
- c) Picking a player or team crest card.
- d) Picking a team crest card.
State whether the following statements about chance are true or false, giving reasons for your answer: a) Getting a $7$ when rolling a die is an impossible event. b) If…
State whether the following statements about chance are true or false, giving reasons for your answer:
- a) Getting a $7$ when rolling a die is an impossible event.
- b) If an event is very unlikely, then it is impossible.
- c) If I flip a coin $10$ times, it is certain that heads will come up exactly $5$ times.
- d) If heads has come up $4$ times in a row, on the next flip tails is more likely "to balance things out".
Comparing events: more or less likely
Between "impossible" and "certain" there are many possible events, and some are easier than others. They are compared with phrases like "is more likely than…", "is less likely than…", or "is equally likely as…".
A practical rule: if the outcomes are equally likely, the more favorable outcomes an event has, the more likely it is.
In a bag with $9$ red marbles and $1$ blue one:
-
"Drawing red" is very likely (9 out of 10 marbles).
-
"Drawing blue" is unlikely (1 out of 10).
-
"Drawing green" is impossible (there are no green ones).
-
"Drawing a marble that isn't green" is certain.
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…
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".
a) Make up a random experiment whose sample space has exactly $6$ equally likely outcomes. Write down its sample space. b) Now make up a random experiment that also has…
- a) Make up a random experiment whose sample space has exactly $6$ equally likely outcomes. Write down its sample space.
- b) Now make up a random experiment that also has $6$ possible outcomes that are not equally likely, and explain why they are not.
- 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 color. a) Describe a certain event for this experiment…
An opaque bag contains only red and blue balls, $10$ balls in total, but we do not know how many there are of each color.
- a) Describe a certain event for this experiment (picking a ball at random).
- b) Describe an impossible event.
- 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.
- d) What if "picking a red ball" were as likely as "picking a blue ball"?
Back to the beginning: Diego and Ximena can't know whether heads or tails will come up because the coin flip is a random experiment: they know its sample space, $E = \{H, T\}$, but the result of each try depends on chance. Since both outcomes are equally likely, the coin flip is a fair way to decide.