Coaching Centre
The language of probability and sample spaces
Words for chance, the probability scale from 0 to 1, writing probabilities as fractions, decimals and percentages, and listing sample spaces for single and multistage experiments.
"There's a good chance the 8:15 to Central will be late." "It's practically impossible to win Powerball." People talk about chance all the time, but words like "good chance" mean different things to different people. Maths fixes that by giving every event a number between 0 and 1. Before you can calculate anything, you need the vocabulary and you need to be able to list everything that could happen.
The words that matter
| Word | Meaning | Example: rolling a die |
|---|---|---|
| Experiment | Something you do that has an uncertain result | Rolling a six-sided die |
| Trial | One go at the experiment | One roll |
| Outcome | One possible result of a trial | Rolling a 4 |
| Sample space | The list of all possible outcomes | $\{1, 2, 3, 4, 5, 6\}$ |
| Event | A collection of outcomes you are interested in | "An even number", which is $\{2, 4, 6\}$ |
The sample space is written in curly brackets, and the number of outcomes in it is often called $n(S)$. An event can be a single outcome or several. "Rolling a number bigger than 4" is the event $\{5, 6\}$, with 2 outcomes.
The probability scale
Every probability sits between 0 and 1. An event that cannot happen has probability 0. An event that must happen has probability 1. Everything else is somewhere in between, and the closer to 1, the more likely.
A probability can be written three ways, and the HSC accepts any of them unless the question asks for one. A one-in-four chance is $\frac{1}{4}$, $0.25$ or $25\%$. To change a fraction to a percentage, multiply by 100: $\frac{3}{8} \times 100 = 37.5\%$.
Read each event and choose the word that fits it best. The marker shows where its exact probability sits on the scale.
Which two events land on exactly the same spot? Why does "picture card" count as unlikely even though there are 12 of them?
Listing a sample space
For one stage, just list. For two stages (toss a coin and roll a die, or spin twice), you need a system so you don't miss anything. A table or a systematic list works. Pair every outcome of the first stage with every outcome of the second, so the total number of outcomes is the two counts multiplied.
A canteen meal deal lets you choose one main (wrap, pie or sushi) and one drink (water or juice). List the sample space and find how many meal deals include a pie.
A spinner has three equal sectors coloured red, blue and green. It is spun twice. List the sample space in a table and find how many outcomes show the same colour twice.
| 1st \ 2nd | R | B | G |
|---|---|---|---|
| R | RR | RB | RG |
| B | BR | BB | BG |
| G | GR | GB | GG |
Your turn
A family has two children. Ignoring twins, list the sample space for the order of boys (B) and girls (G). How many outcomes contain exactly one girl?
Show the answer
$S = \{BB, BG, GB, GG\}$, so $n(S) = 2 \times 2 = 4$. Exactly one girl: BG and GB, which is $2$ outcomes. Students often write $\{BB, BG, GG\}$ and miss that "older boy, younger girl" and "older girl, younger boy" are different.
A die is rolled once. The event "rolling a prime number" is:
A prime has exactly two factors. On a die those are 2, 3 and 5. The number 1 has only one factor, so it is not prime. The event is $\{2, 3, 5\}$.
Which probability describes an event that is likely but not certain?
Likely means more than 0.5 but less than 1. Only $0.8$ fits. $1$ is certain, $0.3$ is unlikely, and $1.2$ is impossible as a probability because probabilities can't exceed 1.
Write $\frac{5}{8}$ as a percentage.
$\frac{5}{8} \times 100 = 62.5$, so $62.5\%$.
The weather bureau gives a $35\%$ chance of rain in Penrith tomorrow. Write this probability as a decimal.
Divide the percentage by 100: $35 \div 100 = 0.35$.
A coin is tossed and a six-sided die is rolled. How many outcomes are in the sample space?
The coin has 2 outcomes and the die has 6. Each coin result pairs with each die result, so $n(S) = 2 \times 6 = 12$.
A spinner with sectors labelled 1, 2 and 3 is spun twice. Which outcome is missing from this list: 11, 12, 13, 21, 22, 23, 31, 32?
There should be $3 \times 3 = 9$ outcomes. Working through systematically, the third row is 31, 32, 33. The missing one is 33.
A bag holds one red, one blue and one yellow counter. One counter is drawn. Which statement is true?
There is no green counter, so $P(\text{green}) = 0$, which is impossible. $P(\text{red}) = \frac{1}{3}$, not certain or $\frac{1}{2}$, and $P(\text{not blue}) = \frac{2}{3}$.
A coin is tossed three times. How many outcomes are in the sample space?
Each toss has 2 outcomes, so $n(S) = 2 \times 2 \times 2 = 8$: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.
Exam style: Jai says, "When I play a game against one other player, I either win, lose or draw, so my chance of winning is $\frac{1}{3}$." Which is the best response?
Dividing by the number of outcomes only works when the outcomes are equally likely. A strong player wins far more than one game in three, and draws may be rare. Without data or equally likely outcomes, $\frac{1}{3}$ is not justified.
Exam style: A Year 12 formal ticket comes with a choice of entrée (soup or prawns), main (chicken, beef or vegetarian) and dessert (cake or fruit). How many of the possible three-course meals include beef?
Fix the main as beef. That leaves 2 entrée choices and 2 dessert choices, so $2 \times 1 \times 2 = 4$ meals include beef (out of $2 \times 3 \times 2 = 12$ meals altogether).