probability

Likelihood, or chance, that an event will occur, often expressed as odds, or in mathematics, numerically as a fraction or decimal.

In general, the probability that n particular events will happen out of a total of m possible events is n/m. A certainty has a probability of 1; an impossibility has a probability of 0.

Probability = number of successful events/total possible number of events

In tossing a coin, the chance that it will land ‘heads’ is the same as the chance that it will land ‘tails’, that is, 1 to 1 or even; mathematically, this probability is expressed as 1/2 or 0.5. An estimate for probability can be achieved by experiment; this is known as the relative frequency. The probability of any chosen number coming up on the roll of a fair die is 5 to 1; the probability is 1/6 or 0.1666. If two dice are rolled there are 6 × 6 = 36 different possible combinations. The probability of a double (two numbers the same) is 6/36 or 1/6 since there are six doubles in the 36 events: (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6).

Independent events are those that do not affect each other, for example rolling two dice are independent events, as the rolling of the first die does not affect the outcome of the rolling of the second die. If events are described as mutually exclusive it means that if one happens, then it prevents the other from happening. So tossing a coin is a mutually exclusive event as it can result in a head or a tail but not both. The sum of the probabilities of mutually exclusive events is always equal to 1. For example, if one has a bag containing 3 marbles, each of a different colour, the probability of selecting each colour would be 1/3.

1/3 + 1/3 + 1/3 = 1

To find out the probability of two or more mutually exclusive events occurring, their individual probabilities are added together. So, in the above example, the probability of selecting either a blue marble or a red marble is

1/3 + 1/3 = 2/3

The probability of two independent events both occurring is smaller than the probability of one such event occurring. For example, the probability of throwing a 3 when rolling a die is 1/6, but the probability of throwing two 3s when rolling two dice is 1/36.

Conditional probability is when the outcome of the first event affects the outcome of the second event. For example, if a ball is chosen at random from a bag of 4 blue balls and 5 red balls, and not replaced, the probability of selecting 2 blue balls is:

P(b) = 4/9 × 3/8 = 12/72 = 1/6

This can be displayed in a tree diagram.

Probability theory was developed by the French mathematicians Blaise Pascal and Pierre de Fermat in the 17th century, initially in response to a request to calculate the odds of being dealt various hands at cards. Today probability plays a major part in the mathematics of atomic theory and finds application in insurance and statistical studies.