Theoretical Probability
Learn to calculate theoretical probability for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-07). When all outcomes are equally likely, an event’s probability is the number of favourable outcomes divided by the total number of outcomes.
You will count the favourable outcomes and the total, apply this rule, and give your answer as a fraction, decimal or percentage — practising with dice, coloured marbles and a standard deck of cards, the kinds of chance you meet in games and raffles.
Theory
Theoretical probability is the number of favourable outcomes divided by the total number of equally likely outcomes. This Year 12 Standard 1 (NSW) guide covers the probability scale from \(0\) to \(1\), dice, marbles and cards, and writing a probability as a fraction, decimal or percentage.
When all the outcomes of an experiment are equally likely, the theoretical probability of an event is the number of favourable outcomes divided by the total number of outcomes.
A probability is a number from \(0\) to \(1\). \(P=0\) means impossible, \(P=1\) means certain, and \(P=\dfrac{1}{2}\) is an even chance. No probability is below \(0\) or above \(1\).
The same probability can be written as a simplified fraction, a decimal (divide the top by the bottom) or a percentage (multiply by \(100\%\)). A standard deck has \(52\) cards: \(4\) suits of \(13\), with \(26\) red cards and \(4\) of each value.
For a finite number of equally likely outcomes:
Every probability lies on the scale from impossible to certain:
How to find a theoretical probability
- Count the total number of outcomes \(n(S)\). Add the groups if they are given separately, e.g. \(4\) red \(+\) \(6\) blue \(=10\) marbles.
- List and count the favourable outcomes \(n(E)\) — the ones that fit the event.
- Divide and simplify \(P=\dfrac{n(E)}{n(S)}\), then give the form asked for: fraction, decimal or percentage.
List the favourable outcomes, then divide by the total.
| \(n(S)\) | \(=\) | \(6\) |
| \(\text{greater than }2\) | \(=\) | \(\{3,4,5,6\}\) |
| \(n(E)\) | \(=\) | \(4\) |
| \(P(\text{greater than }2)\) | \(=\) | \(\dfrac{4}{6}=\dfrac{2}{3}\) |
\(P(\text{greater than }2)=\dfrac{2}{3}\).
Add the marbles for the total, then form favourable over total.
| \(n(S)\) | \(=\) | \(7+5=12\) |
| \(n(\text{yellow})\) | \(=\) | \(5\) |
| \(P(\text{yellow})\) | \(=\) | \(\dfrac{5}{12}\) |
\(P(\text{yellow})=\dfrac{5}{12}\).
| Colour | Red | Blue | White |
|---|---|---|---|
| Number | \(9\) | \(6\) | \(15\) |
Total the tickets, form the fraction, then convert.
| \(n(S)\) | \(=\) | \(9+6+15=30\) |
| \(P(\text{blue})\) | \(=\) | \(\dfrac{6}{30}=\dfrac{1}{5}\) |
| \(\) | \(=\) | \(0.2\) |
| \(\) | \(=\) | \(0.2\times100\%=20\%\) |
\(P(\text{blue})=0.2\), or \(20\%\).
\(n(S)=52\); count each event from the make-up of the deck.
| \(\text{(i) }P(\text{king})\) | \(=\) | \(\dfrac{4}{52}=\dfrac{1}{13}\) |
| \(\text{(ii) }P(\text{diamond})\) | \(=\) | \(\dfrac{13}{52}=\dfrac{1}{4}\) |
| \(\text{(iii) }P(\text{black})\) | \(=\) | \(\dfrac{26}{52}=\dfrac{1}{2}\) |
(i) \(\dfrac{1}{13}\) (ii) \(\dfrac{1}{4}\) (iii) \(\dfrac{1}{2}\)
Common pitfalls
Frequently asked questions
What is theoretical probability?
Theoretical probability is the chance of an event worked out from equally likely outcomes, without doing an experiment. It equals the number of favourable outcomes divided by the total number of outcomes.
How do you calculate probability?
Count the total number of outcomes, count the favourable outcomes, then divide: P = favourable over total. For example, the chance of a 5 or 6 on a die is 2 over 6, which is 1 third.
How do you write a probability as a percentage?
Write the probability as a fraction or decimal, then multiply by 100 percent. For example, 1 fifth is 0.2, and 0.2 times 100 percent is 20 percent.
Can a probability be greater than 1?
No. Every probability is between 0 and 1. A probability of 0 means impossible and 1 means certain, so an answer above 1 means the working has an error.
How many cards are in a standard deck?
A standard deck has 52 cards in 4 suits of 13. Hearts and diamonds are red and clubs and spades are black, so there are 26 red cards, 26 black cards and 4 of each value.
What does an even chance mean?
An even chance means a probability of 1 half, or 0.5, or 50 percent. The event is just as likely to happen as not to happen, like a head when a fair coin is tossed.