Basic Probability

What is probability?

Probability describes the likelihood of something happening
In real-life you might use words such as impossible, unlikely, certain, etc to describe probability
In maths we use the probability scale to describe probability
- This means giving it a number between 0 (impossible) and 1 (certain)
- Probabilities can be given as fractions, decimals or percentages

What key words and terminology are used with probability?

An experiment is a repeatable activity that has a result that can be observed or recorded
- Trials are what we call the repeats of the experiment
An outcome is a possible result of a trial
An event is an outcome or a collection of outcomes
- Events are usually denoted with capital letters: A, B, etc
- n( A) is the number of outcomes that are included in event A
- An event can have one or more than one outcome
A sample space is the set of all possible outcomes of an experiment
- It can be represented as a list or a table
The probability of event A is denoted as P( A)

How do I calculate basic probabilities?

If all outcomes are equally likely then probability for each outcome is the same
- Probability for each outcome is $\frac{1}{Total number of outcomes}$
  - e.g. If there are 50 marbles in a bag then the probability of selecting a specific one is $\frac{1}{50}$
Theoretical probability of an event can be calculated without using an experiment by dividing the number of outcomes of that event by the total number of outcomes

$P (A) = \frac{Total number of outcomes for the event}{Total number of outcomes}$

- Identifying all possible outcomes either as a list or a table can help
  - e.g. If there are 50 marbles in a bag and 20 of then are blue then the probability of selecting a blue marble is $\frac{20}{50}$

How do I find missing probabilities?

The probabilities of all the outcomes add up to 1
- If you have a table of probabilities with one missing then you can find it by subtracting the rest from 1
The complement of event A is the event where event A does not happen
- This can be thought of as not A
- This is denoted A'
  - It is always true that $P (A) + P (A') = 1$

- - This may also be written as $P (A') = 1 - P (A)$

What are mutually exclusive events?

Two events are mutually exclusive if they cannot both happen at once
- For example: when rolling a dice the events “getting a prime number” and “getting a 6” are mutually exclusive
If A and B are mutually exclusive events then to find the probability that A or B occurs you can simply add together the probability of A and the probability of B
Complementary events are mutually exclusive

Exam Tip

Probabilities can be fractions, decimals or percentages (nothing else!). If no format is indicated in the question then fractions are normally best.

Worked example

Emilia is using a spinner that has outcomes and probabilities as shown in the table.

Outcome	Blue	Yellow	Green	Red	Purple
Probability		0.2	0.1		0.4

The spinner has an equal chance of landing on blue or red.

Complete the probability table.

The probabilities of all the outcomes should add up to 1.

1 - 0.2 - 0.1 - 0.4 = 0.3

The probability that it lands on blue or red is 0.3.
As the probabilities of blue and red are equal you can halve this to get each probability.

0.3 ÷ 2 = 0.15

Now complete the table.

Outcome	Blue	Yellow	Green	Red	Purple
Probability	0.15	0.2	0.1	0.15	0.4

Find the probability that the spinner lands on green or purple.

As the spinner can not land on green and purple at the same time they are mutually exclusive.
This means you can add their probabilities together.

0.1 + 0.4 = 0.5

P(Green or Purple) = 0.5

Find the probability that the spinner does not land on yellow.

The probability of not landing on yellow is equal to 1 minus the probability of landing on yellow.

1 - 0.2 = 0.8

P(Not Yellow) = 0.8

Sample Space

What is a sample space?

In probability, the sample space means all the possible outcomes
In simple situations it can be given as a list
- For flipping a coin, the sample space is: Heads, Tails
  - the letters H, T can be used
- For rolling a six-sided die, the sample space is: 1, 2, 3, 4, 5, 6
When combining two things a grid can be used
- For example, rolling two six-sided dice and adding their scores
- A list of all the possibilities would be very long
  - It would be hard to spot whether you had missed any possibilities
  - It would be hard to spot any patterns in the sample space
- Use a grid instead

dice-grid

If you need to combine more than two things you'll probably need to go back to listing
- For example, flipping three coins (or flipping one coin three times!)
  - In this case the sample space is: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT (8 possible outcomes)

How do I use a sample space to calculate probabilities?

Probabilities can be found by counting the possibilities you want, then dividing by the total number of possibilities in the sample space
- For example, in the sample space 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 you can count 4 prime numbers (2, 3, 5 and 7)
  - So the probability of getting a prime number is $\frac{4}{10} = \frac{2}{5}$
- Or for rolling two dice and adding the results, the grid above shows there are 5 ways to get '8', and 36 outcomes in total
  - So the probability of getting an 8 is $\frac{5}{36}$
But be careful - this counting method only works if all possibilities in the sample space are equally likely
- For a fair six-sided die: 1, 2, 3, 4, 5, 6 are all equally likely
- For a fair ( unbiased) coin: H, T are equally likely
- Winning the lottery: Yes, No. These are not equally likely!
  - You cannot count possibilities here to say the probability of winning the lottery is $\frac{1}{2}$
This method can also be used for finding the probability of an event occurring given that another event has occurred ('conditional probability')
- For example when two dice are rolled, you can use the grid above to find the probability that an individual dice shows a 6, given that the total showing on the two dice is 7
  - - count the number of outcomes that sum to 7 (there are 6 of them) - this goes in the denominator
    - count the number of those outcomes in which one of the dice shows a 6 (there are two of these, (1,6) and (6,1)) - this goes in the numerator
    - So the probability is $\frac{2}{6} (= \frac{1}{3})$

Exam Tip

Some harder questions may not say "by drawing a grid" so you have to decide to do it on your own.

Worked example

Two fair six-sided dice are rolled.

Find the probability that the sum of the numbers showing on the two dice is an odd number greater than 5, giving your answer as a fraction in simplest form.

Draw a sample space grid to show all the possible outcomes.

Circle the possibilities that are odd numbers greater than 5.
(5 is not included!)

C ount the number of possibilities that are circled (12) and divide them by the total number of possibilities in the grid (36).

\frac{12}{36}

Finally, simplify your answer.

\frac{12}{36} = \frac{12 \times 1}{12 \times 3} = \frac{1}{3}

\frac{1}{3}

Given that the sum of the numbers showing on the two dice is an odd number greater than 5, find the probability that one of the dice shows the number 2. Give your answer as a fraction in simplest form.

From part (a) you already know there are 12 ways to get an odd number greater than 5.
Now find how many of those ways (i.e., how many of the circled possibilities in the grid) involve one of the dice showing the number 2.
There are two of these: (2, 5) and (5, 2).
So the probability we are looking for is 2 divided by 12.

\frac{2}{12}

Finally, simplify your answer.

\frac{2}{12} = \frac{2 \times 1}{2 \times 6} = \frac{1}{6}

\frac{1}{6}

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