Probability is an interesting branch of mathematics that is widely used in genetics, insurance, finance, medicine, sociological surveys, marketing and science. We use probability to measure the chance or likelihood of an event or events occurring in the future.
Events
An event is something that may or may not occur at some time or during some period in the future.
When we talk about events in terms of chances, we could make statements such as:
"I will probably play tennis this summer"
"It isn't likely that I will be invited to play in the Australian Open Tennis Tournament"
"My chances of winning Tattslotto are not very good"
We could describe the expected occurrence of an event (or events) with the words certain, probable, fifty-fifty, improbable and impossible. These words tell us something about whether or not the event is expected to occur; but the statements are very vague. We study and apply probability to enable us to better quantify or measure the chance of an event (or events) occurring in the future.
The Probability Scale
If an event is impossible, its probability is 0. If an event is certain to occur, its probability is 1. The probability of any other event is between these two values. That is:
Probability Experiment
A probability experiment involves performing a number of trials to enable us to measure the chance of an event occurring in the future. A trial is a process by which an outcome is noted.
For example, consider the following three experiments:
Experiment: Roll a die two hundred times noting the outcomes.
Event of interest: A six faces upwards.
Trial: Roll the die once.
Number of trials: 200
Outcomes: 1, 2, 3, 4, 5 or 6
Experiment: Toss a coin seventy times noting the outcomes.
Event of interest: A tail faces upwards.
Trial: Toss the coin once.
Number of trials: 70
Outcomes: Head or Tail
Experiment: Spin a spinner one hundred times noting the outcomes.
Event of interest: The spinner stops on the number 3.
Trial: Spin the spinner once.
Number of trials: 100
Outcomes: 1, 2, 3, 4
The Sample Space
The sample space of an experiment is the set of all possible outcomes of any trial of the experiment to be conducted.
For example, if an unbiased coin is tossed, then the two possible outcomes are 'head' and 'tail'. The set of all possible outcomes is therefore {H, T}. This is called the sample space of the experiment and is denoted by S.
An element of the sample space is called a sample point.
For the example under consideration, the sample points are H and T.
Activity 16.1
To investigate the probability of obtaining a head (or tail) when a coin is tossed.
Experiment - Toss the coin 100 times and complete the following table.
| Outcome |
Tally of outcomes
| Total number of trials in which the outcome occurred |
Tail
Head | | |
Sum
| |
If the coin is unbiased (i.e. well balanced), we say it is just as likely to turn up heads as tails. In the activity under consideration, we would expect about half of the coin tosses to turn up heads. Does this agree with your result?
Long Run Proportion
Long run proportion is defined as the ratio of favourable outcomes to the total number of trials in an experiment after conducting a very large number of trials.
When we express the 50 chances of tossing a head (or tail) to the 100 coin tosses as a ratio, we are describing the chances of tossing a head (or tail) in terms of a probability or long run proportion.
From the preceding discussion, we can conclude that:
The concept of long run proportion enables us to determine the probability (or chance) of an event.
We use the data collected in surveys to make predictions.
Example 1
A survey was conducted using 200 randomly selected students from across the state to investigate which of 5 sports is played more often? The question put to each student was which sport out of basketball, cricket, football, netball and tennis do you play more often? The tabulated results appear below.
Sport
|
Number of students
|
Basketball
Cricket
Football
Netball
Tennis
|
34
10
44
60
52
|
Sum
|
200
|
If the results give a fair indication of how all students in the state would have answered the survey question, calculate the probability that any randomly selected student from the state plays the following sport the most?
a. Basketball
b. Cricket
c. Football
d. Netball
e. Tennis
Solution:
That is, the sum of the probabilities of all events is equal to 1.
Recall that:
The sample space (S) of an experiment is the set of all possible outcomes of any trial of the experiment to be conducted.
An event (E) is a subset of the sample space. That is, an event is a subset of all possible outcomes. We refer to this subset of outcomes as favourable outcomes.
The probability of event E occurring is given by
This is often written as:
This result holds only if the outcomes of an experiment are equally likely.
Note:
The events are denoted by capital letters A, B, C, D, E, ...
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Example 2
A die is rolled. Find:
a. the sample space for this experiment
b. the probability of obtaining a number less than 4
c. the probability of obtaining a square number |
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Solution:
Example 3
A die is rolled. Find the probability of obtaining:
a. a 7
b. a number less than or equal to 6
Solution:
Note:
- It is certain that event A will occur as it contains all 6 possible outcomes.
- 7 is not an outcome of rolling a die as it is not possible.
Example 4
A pack of 52 playing cards consists of four suits, i.e. clubs, spades, diamonds and hearts. Each suit has 13 cards which are the 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king and the ace card. Clubs and spades are of black colour whereas diamonds and hearts are of red colour. So, there are 26 red cards and 26 black cards.
Find the probability of drawing from a well-shuffled pack of cards:
a. a black card
b. the queen of diamonds
c. a king
Solution:
 
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