Relative Frequency and Probability

Question 1

Language of probability

Answer 1

Probability language is used to specify the chances of an event occurring. For example:
‘The chance of rain next week is 50%.’
Events that must occur are certain and events that have no chance of occurring are impossible. When an event has an equal chance of occurring or not occurring, it is an even chance or 50−50 or a 50% chance. Events that have better than an even chance are called likely, often, probable, sure or expected. Conversely, events that have less than an even chance are called unlikely, doubtful, improbable, rarely or unexpected.

Question 2

Probability - formula

Answer 2

Probability (Event) = Number of favourable outcomes OVER Total number of outcomes

Question 3

Sample space

Answer 3

Sample space is the set of all possible outcomes or possible results of an experiment. For example, if the experiment is tossing a coin, then the sample space is a head and a tail. When a die is tossed, the sample space is the numbers 1, 2, 3, 4, 5 and 6. Each outcome or data value is an element of the sample space. The sample space is usually listed between curly brackets {}.

Question 4

Tables

Answer 4

A table is an arrangement of information in rows and columns. The table shows all the possible outcomes for tossing two coins. There are two events: tossing the first coin and tossing the second coin. The outcomes of the first event are listed down the first column (Head or Tail). The outcomes of the second event are listed across the top row (Head or Tail). Each cell in the table is an outcome. There are four possible outcomes.
Sample space = {HH, HT, TH, TT}

Question 5

Tree diagram

Answer 5

A tree diagram is a diagram that details the outcomes of a multistage experiment. It shows each event as a branch of the tree. The tree diagram below shows all the possible outcomes for tossing two coins. The outcomes of the first event are listed (H or T) with two branches. The outcomes of the second event are listed (H or T) with two branches on each of the outcomes from the first event. The sample space is HH, HT, TH and TT.

Question 6

Range of probabilities

Answer 6

The probability of an event that is impossible is 0 and the probability of an event that is certain is 1. Probability is always within this range, or from 0 to 1. It is not possible to have the probability of an event as 2. The range of probability is expressed as 0 ≤ P(E) ≤ 1 or P(E) ≥ 0 and P(E) ≤ 1. It is also important to realise that the probability of every event in an experiment will sum to 1.

Question 7

Complementary events

Answer 7

The complement of an event E is the event not including E. For example, when throwing a die the complement of 2 are the events 1, 3, 4, 5 and 6. The complement of an event E is denoted by E. An event and its complement represent all the possible outcomes and are certain to occur. Hence the probability of an event and its complement will sum to be 1.

Question 8

Relative frequency

Answer 8

Relative frequency is calculated when an experiment is performed. The frequency of an event is the number of times the event occurred in the experiment. Relative frequency is the frequency of the event divided by the total number of frequencies. It is also known as experimental probability, as it estimates the chances of something happening or the probability of an event. Relative frequency is expressed using fractions, decimals and percentages.

Question 9

Relative frequency - formula

Answer 9

Relative frequency = Frequency of the data value OVER Total number of data values

Question 10

Expected frequency

Answer 10

Expected frequency is the number of times that a particular event should occur.
Expected frequency = n × p = np
n – number of times the experiment is repeated
p – probability of the event