Year 10 Interactive Maths - Second Edition

## Probability

Probability is a 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 probably, likely or chances.  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.

### Probability Experiment

A probability experiment is a test in which we perform a number of trials to enable us to measure the chance of an event occurring in the future.  The results from a probability experiment may reveal a known truth or lead us to discover something about the probability (or chance) of an event occurring in the future.

A trial is a process by which an outcome is noted.

Throwing a die, tossing a coin, rotating a spinner and drawing a card from a pack of playing cards are all examples of probability experiments.

Note that a trial produces one and only one outcome from all the possible outcomes.

### 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

Investigate the probability (or chance) of obtaining a diamonds card from a well-shuffled pack of 52 playing cards.

Experiment - Draw one card and record, in the tally column, whether it is a diamonds, hearts, clubs or spades card. Return the card to the pack, and re-shuffle it. Repeat this process 156 times and complete the following table.

 Outcome Tally of outcomes Total number of trials in which the outcome occurred Diamonds Hearts Clubs Spades Sum

In the activity under consideration, we would expect 39 of the 156 drawn cards to be diamonds.  Does this agree with your result?

When we express the 39 chances of drawing a diamonds card to the 156 drawn cards as a ratio, we are describing the chances of drawing a diamonds card in terms of a probability.

### 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.

###### 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.