## Probability

Probability is a measure of how likely it is that an event will happen.

### Definitions

• The rolling of a die, the drawing of a card, etc. are called experiments.
• trial is a single part of an experiment which consists of many trials.

e.g. Rolling the die 10 times would be 10 trials.

• An outcome is the result of a trial of an experiment.
• The sample space (S) is the set of all the possible outcomes of an experiment.
• An event (E) is a part of the sample space.

 e.g. An experiment could be the tossing of two coins. The sample space for this experiment is: S = {(head, head), (head, tail), (tail, head), (tail, tail)} An event could be: E = {both the coins are the same} = {(head, head), (tail, tail)}

### Probability

Probabilities are expressed as fractions, decimal fractions or percentages.

For equally likely outcomes, the theoretical probability of the event E occurring is given by:

P(E) =

For experiments, the experimental probability of an event E happening is given by:

P(E) =

For experiments, the more trials that are carried out, the nearer the experimental
probability will be to the theoretical probability given above.

This can be shown using an experiment with a spinner −
(This activity was created by the Shodor Education Foundation.)

 Example 1 Answer When two coins are tossed together, what is the probability that they will both be the same? S = {(H, H), (H, T),(T, H), (T, T)} E = {(H, H), (T, T)} Example 2 Answer If a die is rolled, what is the probabilityof getting more than a 4? S = {1, 2, 3, 4, 5, 6} E = {5, 6}

Probabilities must always be in the range from 0 to 1.

If the probability of an event happening is 0, the event cannot occur.

If the probability of an event happening is 1, the event is certain to occur.

e.g. P(throwing a six-sided die and scoring a 7) = 0

P(throwing a six-sided die and scoring less than 7) = 1

See an activity involving probabilities when throwing two dice − .
(This activity was created by the Shodor Education Foundation.)

Multiplication Principle

To find the probability of multiple events occurring, use the multiplication principle and multipy the individual probabilities together.

e.g. P(throwing a "6" on a standard dice followed by a "5")

= P(throwing a "6") x P(throwing a "5")

=

### Expected value

The expected value of an event occurring from a certain number of trials is the number of times the event is expected to occur.

Expected value = (Probability of event) x (number of trials)

= P(E) x n

 Example Answer If a die is thrown 60 times, how many times would you expect a 4 or a 5 to be thrown?