Probability Basics
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
 This is denoted by U
 n(U) is the total number of outcomes
 It can be represented as a list or a table
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
 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

 This is given in the formula booklet
 Experimental probability (also known as relative frequency) of an outcome can be calculated using results from an experiment by dividing its frequency by the number of trials
 Relative frequency of an outcome is
How do I calculate the expected number of occurrences of an outcome?
 Theoretical probability can be used to calculate the expected number of occurrences of an outcome from n trials
 If the probability of an outcome is p and there are n trials then:
 The expected number of occurrences is np
 This does not mean that there will exactly np occurrences
 If the experiment is repeated multiple times then we expect the number of occurrences to average out to be np
What is the complement of an event?
 The probabilities of all the outcomes add up to 1
 Complementary events are when there are two events and exactly one of them will occur
 One event has to occur but both events can not occur at the same time
 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'


 This is in the formula booklet
 It is commonly written as

Worked Example
Dave has two fair spinners, A and B. Spinner A has three sides numbered 1, 4, 9 and spinner B has four sides numbered 2, 3, 5, 7. Dave spins both spinners and forms a twodigit number by using the spinner A for the first digit and spinner B for the second digit.
is the event that the twodigit number is a multiple of 3.
a)
List all the possible twodigit numbers.
b)
Find .
c)
Find .
Independent & Mutually Exclusive Events
What are different types of combined events?
 The intersection of two events (A and B) is the event where both A and B occur
 This can be thought of as A and B
 This is denoted as
 The union of two events (A and B) is the event where A or B or both occur
 This can be thought of as A or B
 This is denoted
 The event where A occurs given that event B has occurred is called conditional probability
 This can be thought as A given B
 This is denoted
What are mutually exclusive events?
 Two events are mutually exclusive if they can not 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:
 Complementary events are mutually exclusive
What are independent events?
 Two events are independent if one occurring does not affect the probability of the other occurring
 For example: when flipping a coin twice the events “getting a tails on the first flip” and “getting a tails on the second flip” are independent
 If A and B are independent events then:
 and
 If A and B are independent events then:

 This is given in the formula booklet

How do I find the probability of combined events?
 The probability of A or B (or both) occurring can be found using the formula


 This is given in the formula booklet
 You subtract the probability of A and B both occurring because it has been included twice (once in P(A) and once in P(B) )

 If A and B are mutually exclusive events then


 This is given in the formula booklet
 This occurs because

 For any two events A and B the events and are mutually exclusive and A is the union of these two events

 This works for any two events A and B

Worked Example
a)
A student is chosen at random from a class. The probability that they have a dog is 0.8, the probability they have a cat is 0.6 and the probability that they have a cat or a dog is 0.9.
Find the probability that the student has both a dog and a cat.
Find the probability that the student has both a dog and a cat.
b)
Two events, and , are such that and .
Given that and are independent, find .
Given that and are independent, find .
c)
Two events, and , are such that .
Given that and are mutually exclusive and that find and .
Given that and are mutually exclusive and that find and .