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Uncertainty is a feature of everyday life. Probability is an area of maths that addresses how likely things are to happen.
A good understanding of probability is important in many areas of work. It is used by scientists, governments, businesses, insurance companies, betting companies and many others, to help them anticipate future events.
A statistics experiment will have a number of different outcomes. The set of all possible outcomes is called the sample space of the experiment.
Introduction to probability
An event is a collection of some of the outcomes from an experiment. For example, getting an even number on the dice or scoring more than 40 on the quiz.
In a general knowledge quiz with 70 questions, the sample space for the number of questions a
person answers correctly is {0, 1, 2, …, 70}.
if a normal dice is thrown the sample space would be {1, 2, 3, 4, 5, 6}.
Some probabilities are less simple. It is not always possible to calculate how likely each outcome is.
Estimating probability
number of times event occursnumber of times experiment is repeated
You can increase the accuracy of the relative frequency as an estimate of probability, by increasing
the number of times you repeat the experiment.
This is referred to as the relative frequency.
However, the probability of an event happening can be estimated experimentally, by repeating an experiment over and over again. The probability is estimated using:
Two events A and B are called mutually exclusive if they cannot occur at the same time.
A BIf A and B are mutually exclusive, then:
Addition properties
This symbol means ‘union’ or ‘OR’
However the events “the card is a club” and “the card is a queen” are not mutually exclusive.
For example, if a card is picked at random from a standard pack of 52 cards, the events “the card is a club” and “the card is a diamond” are mutually exclusive.
Tree diagrams are sometimes a useful way of finding probabilities that involve a succession of events.
Independent events
Example: A bag contains 6 green counters and 4 blue counters. A counter is chosen at random from the bag and then replaced. This is repeated two more times.Find the probability that the 3 counters chosen are:
However, the probability of event B happening might depend on whether A has happened or not.
For example, if blue and green counters are pulled from a bag twice and not replaced, then the probability of pulling out a green counter on the second try will depend on what colour was pulled out on the first try.
Conditional probability
The probability that event A will happen, given that event B has happened, is written
Example: A bag contains 8 dark chocolates and 4 milk chocolates. One chocolate is taken out and eaten. A second chocolate is then taken. Find the probability that: