Sample spaces, counting and relative frequency
Seán waits for the bus every morning. Some days it is late, some days it is on time. Can we put a number on "how likely is it that it's late two days in a row?" By the end of this page you will be able to, and you will see how that number links to what actually happens over many mornings.
Outcomes and sample spaces
An outcome is one possible result of an experiment. The sample space is the list of all possible outcomes. An event is a set of outcomes we are interested in, such as "a number greater than 4".
Rolling a fair six-sided die: the sample space is $\{1, 2, 3, 4, 5, 6\}$. There are 6 outcomes and they are all equally likely (the die is fair).
Why does this work? If there are 6 equally likely outcomes, each gets a "share" of $\frac{1}{6}$ of the certainty. An event with 2 outcomes gets 2 shares.
A bag has 5 red, 3 blue and 2 green counters (10 in total). The event "not blue" has $10 - 3 = 7$ outcomes, so $P(\text{not blue}) = \frac{7}{10}$. This is the complement rule:
$$P(\text{not } A) = 1 - P(A)$$A fair six-sided die is rolled once. a) List the sample space. b) Find the probability of rolling a number greater than 4. c) Find the probability of rolling a prime…
A fair six-sided die is rolled once.
- a) List the sample space.
- b) Find the probability of rolling a number greater than 4.
- c) Find the probability of rolling a prime number.
- d) Find the probability of rolling a 7, and state what this tells you about the event.
A bag contains 5 red, 3 blue and 2 green counters. One counter is picked at random. a) Find the probability that it is red. b) Find the probability that it is not blue.…
A bag contains 5 red, 3 blue and 2 green counters. One counter is picked at random.
- a) Find the probability that it is red.
- b) Find the probability that it is not blue.
- c) Find the probability that it is red or green.
Two-way tables and the counting principle
When two things happen (two dice, say) we can list the outcomes in a two-way table. Here are two dice, with the sum in each cell.
| + | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
There are $6 \times 6 = 36$ equally likely outcomes. The sum 7 appears 6 times, so $P(\text{sum is } 7) = \frac{6}{36} = \frac{1}{6}$. The sum 2 appears once: $\frac{1}{36}$.
Drawing every outcome gets slow when there are many. Use the counting principle.
Two fair dice are rolled and their scores are added. a) Draw a two-way table showing all possible outcomes. b) Find the probability that the sum is 8. c) Find the…
Two fair dice are rolled and their scores are added.
- a) Draw a two-way table showing all possible outcomes.
- b) Find the probability that the sum is 8.
- c) Find the probability that the sum is at least 10.
- d) Find the probability that the sum is a multiple of 5.
A café in Cork offers 4 types of sandwich, 3 types of soup and 5 types of drink. A lunch deal consists of one sandwich, one soup and one drink. a) How many different…
A café in Cork offers 4 types of sandwich, 3 types of soup and 5 types of drink. A lunch deal consists of one sandwich, one soup and one drink.
- a) How many different lunch deals are possible?
- b) Aoife does not want soup. How many different sandwich-and-drink combinations can she choose?
- c) Two of the drinks are hot. How many lunch deals include a hot drink?
A car registration plate is to be made from 2 different letters chosen from $\{A, B, C, D, E\}$ followed by a 3-digit number using digits 1 to 9, where digits may…
A car registration plate is to be made from 2 different letters chosen from $\{A, B, C, D, E\}$ followed by a 3-digit number using digits 1 to 9, where digits may repeat.
- a) How many different plates are possible?
- b) How many plates have an even last digit?
- c) A plate is chosen at random. Find the probability that it contains the letter $A$.
Tree diagrams
A tree diagram shows events happening one after another. Each branch has its probability written on it.
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Multiply along the branches to get the probability of a full path.
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Add the probabilities of the paths that make up the event you want.
Seán's bus is late with probability $0.15$, so it is not late with probability $1 - 0.15 = 0.85$. Over two independent mornings:
Without replacement
Suppose a bag has 4 red and 6 white discs and you draw two without replacement (you keep the first disc out). After the first draw there are only 9 discs left, so the second probabilities change.
If the first disc is red: $P(\text{red}) = \frac{3}{9}$, $P(\text{white}) = \frac{6}{9}$.
If the first disc is white: $P(\text{red}) = \frac{4}{9}$, $P(\text{white}) = \frac{5}{9}$.
For example $P(RR) = \frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15}$.
With replacement, the bag is back to normal for the second draw, so the second-stage probabilities are the same as the first and the events are independent.
The probability that Seán's bus is late on any given morning is $0.15$. The probability that it is early is $0.1$. a) Find the probability that the bus is on time. b)…
The probability that Seán's bus is late on any given morning is $0.15$. The probability that it is early is $0.1$.
- a) Find the probability that the bus is on time.
- b) Draw a tree diagram for two consecutive mornings, assuming the bus is either late or not late and mornings are independent.
- c) Find the probability that the bus is late on exactly one of the two mornings.
A spinner has three equal sectors labelled 1, 2 and 3. It is spun twice, and a coin is tossed once. a) How many outcomes are in the sample space? b) Find the probability…
A spinner has three equal sectors labelled 1, 2 and 3. It is spun twice, and a coin is tossed once.
- a) How many outcomes are in the sample space?
- b) Find the probability that the two spins give the same number and the coin shows heads.
- c) Find the probability that the sum of the two spins is 4 or more, and the coin shows tails.
A bag contains 4 red and 6 white discs. Two discs are drawn at random one after the other, without replacement. a) Draw a tree diagram showing all outcomes and their…
A bag contains 4 red and 6 white discs. Two discs are drawn at random one after the other, without replacement.
- a) Draw a tree diagram showing all outcomes and their probabilities.
- b) Find the probability that the two discs are the same colour.
- c) Find the probability that at least one disc is red.
- d) Explain how the answers would change if the first disc were replaced before the second draw.
Relative frequency
Sometimes we cannot list equally likely outcomes, or we want to test whether a coin really is fair. We run an experiment and record what happens.
A coin is tossed 200 times and lands on heads 94 times. The relative frequency of heads is $\frac{94}{200} = 0.47$. For tails: $\frac{106}{200} = 0.53$. The two add to 1.
The more trials you do, the closer relative frequency usually gets to the true probability.
Use the widget to flip a coin many times and watch the relative frequency settle near $0.5$.
If a coin is fair, you expect about $P \times n$ heads in $n$ tosses. In 1,000 tosses: $0.5 \times 1000 = 500$ heads. Expect is not a promise. Real results vary a little.
A coin is tossed 200 times and lands on heads 94 times. a) Write the relative frequency of heads as a decimal. b) Write the relative frequency of tails as a decimal. c)…
A coin is tossed 200 times and lands on heads 94 times.
- a) Write the relative frequency of heads as a decimal.
- b) Write the relative frequency of tails as a decimal.
- c) How many heads would you expect in 1,000 tosses of a fair coin?
Back to the start: the chance that Seán's bus is late two mornings running is $0.15 \times 0.15 = 0.0225$, about 2 mornings in every 100 pairs. Relative frequency over many mornings would settle near this value.