Probability

These QuickNotes give you the most important points to remember for the ESSLCE. They are based on the MoE Grade 9 Mathematics Textbook, Unit 9, Chapter 9.2.

~5 min read

Summary

Probability is a number between 0 and 1 that describes how likely an event is to happen.
The sample space S is the set of all possible outcomes of an experiment, and an event is a subset of the sample space.
Experimental probability comes from repeating a trial many times. Theoretical probability is P(E) = n(E) / n(S) when all outcomes are equally likely.
A tree diagram lists every outcome of a repeated experiment, such as the 8 outcomes of tossing a coin three times.

Key Words

Experiment: a trial whose outcome cannot be predicted in advance, such as tossing a coin.
Outcome: one possible result of an experiment.
Sample space (S): the set of all possible outcomes of an experiment.
Event: a subset of the sample space, that is, a collection of outcomes you are interested in.
Equally likely: outcomes that occur equally often when the experiment is repeated many times.
Experimental probability: a probability found from data collected by repeating an experiment.
Theoretical probability: the probability P(E) = n(E) / n(S), found by counting outcomes instead of experimenting.
Tree diagram: a branching picture that shows all possible outcomes of a repeated experiment.

What Is Probability?

Probability is a numerical value that describes the likelihood that an event will occur in an experiment.
An experiment is a trial whose outcome you cannot predict in advance. When you toss a fair coin, nobody can say beforehand whether it will land heads or tails.
Every probability is a number from 0 to 1. An impossible event has probability 0, a certain event has probability 1, and an even chance is 1/2.
For example, when you throw one die, getting the number 8 is impossible, but getting a natural number is certain.

Sample Space and Events

The sample space S is the set of all possible outcomes of an experiment.
Tossing one coin gives S = {H, T}. Throwing one die gives S = {1, 2, 3, 4, 5, 6}. Tossing two coins gives S = {HH, HT, TH, TT}.
An event is a subset of the sample space. For example, when throwing a die, “getting an odd number” is the event E = {1, 3, 5}.
A certain event contains every outcome, so its probability is 1. An impossible event contains no outcome, so its probability is 0.

Worked example: the numbers 4 to 30 are written on 27 identical cards and one card is chosen at random. List the elements of some events.

The sample space is S = {4, 5, 6, …, 30}.

The event “the number is greater than 15” is {16, 17, 18, …, 30}.

The event “the number is divisible by 3” is {6, 9, 12, …, 30}.

The event “the number is even” is {4, 6, 8, …, 30}.

Experimental Probability

Experimental probability is found from data collected by repeating an experiment many times.
It equals the number of times the event happened divided by the total number of trials.
Different runs of the experiment can give slightly different values. When you repeat the experiment a very large number of times, the result gets closer and closer to the theoretical probability.

Worked example: a coin is tossed 10,000 times and shows heads 5,010 times. How many tails appeared, and what is the experimental probability of tails?

Tails appeared 10,000 − 5,010 = 4,990 times.

P(tails) = 4,990 / 10,000 = 0.499, which is very close to the theoretical value 0.5.

Another example: a supervisor chose workers at random 100 times and got a man 45 times. The experimental probability of selecting a man is 45/100.

Tree Diagrams

A tree diagram shows all possible outcomes of a repeated experiment as branches.
Each toss of a coin doubles the number of branches. Three tosses give 2 × 2 × 2 = 8 outcomes.
To find a probability from the tree, count the outcomes that match your event and divide by the total number of outcomes.

Worked example: three coins are tossed. Use the tree diagram to find some probabilities.

The sample space is S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}, so there are 8 outcomes.

P(exactly two heads) = 3/8, from the outcomes HHT, HTH and THH.

P(exactly one tail) = 3/8, because exactly one tail means exactly two heads.

P(three tails) = 1/8, from the single outcome TTT.

P(at least two heads) = 4/8 = 1/2, from HHH, HHT, HTH and THH. “At least two” includes exactly two and also all three.

Theoretical Probability

It is not always possible to repeat an experiment, so we also calculate probability by counting.
When all outcomes are equally likely, the theoretical probability of an event E is the number of favourable outcomes divided by the total number of outcomes.

P(E) = n(E) / n(S)

Worked example: you throw a die once. What is the probability that an odd number shows on the upper face?

S = {1, 2, 3, 4, 5, 6} and the event is E = {1, 3, 5}.

P(E) = n(E) / n(S) = 3/6 = 1/2.

Worked example: a coin and a die are thrown together. Find the probability of getting a tail and an odd number.

The combined sample space has 2 × 6 = 12 outcomes, from (H, 1) up to (T, 6).

A tail with an odd number happens in (T, 1), (T, 3) and (T, 5), so P = 3/12 = 1/4.

In the same way, a head with a number less than or equal to 5 happens in 5 outcomes, so P = 5/12.

Question	Experimental probability	Theoretical probability
Where does it come from?	Data from repeating the experiment	Counting equally likely outcomes
Formula	Times the event happened / total trials	n(E) / n(S)
Example	4,990 tails in 10,000 tosses gives 0.499	P(tail) = 1/2 for a fair coin
Do they agree?	With many repeats, the experimental value gets close to the theoretical value.

Common Mistakes to Avoid

Treating HT and TH as the same outcome. The order matters, so two coins give 4 outcomes, not 3.
Reading “at least two heads” as “exactly two heads”. At least two means two or more, so it also includes three heads.
Giving a probability greater than 1 or less than 0. Such an answer is always wrong, so use this as a quick check.
Counting the sample space of a combined experiment wrongly. A coin and a die together give 2 × 6 = 12 outcomes, not 8.
Forgetting to count the whole sample space before dividing. Always find n(S) first, then n(E).

Easy Ways to Remember

Probability is part over whole: favourable outcomes over total outcomes.
0 means never, 1 means sure, and 1/2 means an even chance.
To count outcomes of repeated trials, multiply the choices: 2 × 2 × 2 = 8 for three coins.
“At least” means that many or more. “Exactly” means that many and no more.

Quiz

Tap an answer to check it.

1. Two coins are tossed together. What is the probability of getting exactly one head?

2. A fair die is rolled once. What is the probability that a 1, 4, 5 or 6 shows on the upper face?

3. A bag contains 5 red balls, 2 blue balls and 3 black balls. If you pick one ball at random, what is the probability of getting a black ball?

4. Three fair coins are tossed at the same time. What is the probability of getting three tails?

5. One die is thrown once. Which of the following events is impossible?

Remember: Probability is a number from 0 (impossible) to 1 (certain). List the sample space first, then count the favourable outcomes: P(E) = n(E) / n(S). A tree diagram is the safest way to list every outcome of a repeated experiment.