Random Experiments and Sample Space
In the neighborhood raffle, nobody knows who will be drawn first. When you flip a coin before a baseball game at the park, you don't know whether it will land heads or tails either. That has a name in math: chance. Can we say anything for sure about something we can't predict? Yes: we can know which outcomes are possible and how many there are. Let's get started.
Random experiments and sample space
A sample space isn't always the numbers from 1 to $n$. If you record the suit of a card, $E$ contains words; if you record a team's goals, $E = \{0, 1, 2, 3, \dots\}$.
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 color of the first car that passes by in the street.
e) Calculating the price of $3$ kg of oranges at $\text{RD\$}\ 144$ 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 an outcome or a set of outcomes of the experiment (some books also call it an occurrence).
| 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" |
Between "possible" and "certain" there are degrees: if a bag has 6 red balls and 1 green one, drawing red is very likely; drawing green is unlikely. Both are still possible.
Events are also classified by their size:
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Simple event: has just one outcome (on a die, "rolling a 5").
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Compound event: has two or more outcomes (on a die, "rolling an even number" = $\{2, 4, 6\}$).
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.
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…
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…
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.
Equally likely outcomes and a first idea of probability
Two outcomes are equally likely when they have the same chance of occurring. On a fair die, all six faces are equally likely. On a spinner with sectors of different sizes, they are not: the color with the biggest sector is the most likely, and equal sectors tie.
Careful: just because an experiment has 3 outcomes doesn't mean they are equally likely. When you flip two coins, $\{HH, HT, TT\}$ looks complete, but heads-tails and tails-heads are two different paths, so the outcome "one head and one tail" happens twice as often as the others.
When the cases are equally likely, the probability of an event measures how likely it is and is calculated like this:
$$P(\text{event}) = \frac{\text{favorable cases}}{\text{possible cases}}$$Favorable cases are the ones that satisfy the event; possible cases are all the equally likely cases. This rule is studied in depth in the section on Laplace's rule; here we use it to compare events.
Look at the spinner in the figure. a) Which color is most likely to come up when the pointer is spun? And which is least likely? b) Are there colors with the same…
Look at the spinner in the figure.
a) Which color is most likely to come up when the pointer is spun? And which is least likely?
b) Are there colors with the same probability? Which ones?
c) Order the colors from most likely to least likely.
Compound experiments: two-way table
When an experiment has several stages (two dice, a coin and a die…), a two-way table is used so you don't miss any outcome: one row for each outcome of the first stage and one column for each outcome of the second. Each cell is an outcome.
Roll the two dice many times and notice which sums 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.
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…
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.
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 $1$.
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…
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?
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…
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?
Tree diagrams
When there are two, three, or more stages with few options each, the tree diagram is very clear: each branch is a possibility and each complete path, from the root to the tip, is an outcome.
Multiplication principle: if one stage has $m$ options and another has $n$, there are $m \times n$ outcomes in total. For example, 3 shirts, 2 pairs of pants, and 2 pairs of sneakers make $3 \times 2 \times 2 = 12$ outfits. A 3-wheel code with 4 symbols on each wheel gives $4 \times 4 \times 4 = 64$ codes.
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 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?
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…
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 colors different?
c) In how many outcomes are there at least two identical colors?
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…
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'?
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?
For training, a basketball club has $3$ shirts (red, blue and green), $2$ pairs of shorts (black and white) and $2$ pairs of sneakers (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 sneakers (model A and model B).
a) Draw a tree diagram and calculate how many different uniforms can be formed.
b) How many uniforms include the red shirt?
c) How many include the black shorts and model A sneakers?
The code for a locker at the sports center 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 center 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…
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;…
Compare these two experiments with urns:
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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.
Back to the beginning: you don't know what will come up in a raffle or with a coin, but you can list all the possible outcomes. With two dice, the most likely sum is $7$, with $6$ of the $36$ cases.