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e LEARNING - MATHEMATICS:

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# TOPIC: PROBABILITY

PROBABILITY DEFINITION

Life is full of various events; some are certain; some are impossible; and some are likely. The concept of probability in mathematics deals with the third type of events. We are neither interested in certain events nor impossible events because their outcomes are known already. For instance, if today were to be Monday, it is certain that tomorrow will be Tuesday. Nothing can change that; there is no point in trying to measure the likelihood or calculating probability of that happening. Also, if today were to be Monday, it is impossible that tomorrow will be Thursday. So, there is no need to measure the chances of that occurring. However, if an event is likely to occur, we can measure the chances of that happening - that is what probability is about. Therefore, probability can be defined as the likelihood or chances of an event occurring. Some probability examples include: the probability that it will rain tomorrow, probability the stock market will fall tomorrow, probability of obtaining a score of 4 when a die is rolled, probability of picking a queen from a pack of playing cards etc.

MEANING OF PROBABILITY TERMS

EVENTS: These are the outcomes of experiments.

EXPERIMENT: This is an act that can be repeated under a given circumstance.

OUTCOME: This is the result of random experiment.

RANDOM EXPERIMENT: This is an experiment whose outcome cannot be predetermined.

MUTUALLY EXCLUSIVE EVENTS: These are events that cannot occur simultaneously. This implies that the occurrence of one event automatically means the non-occurrence of the other.

INDEPENDENT EVENTS: As the name implies, these are events that can occur separately or simultaneously. This implies that the occurrence or non-occurrence of one event will not affect the other.

DEPENDENT EVENTS: These are events in which the occurrence or non-occurrence of one event will affect the other event.

EQUALLY LIKELY: Two or more events are said to be equally likely when there is the same chance of their occurrence.

SAMPLE SPACE: This is all probable outcomes of an experiment.

SAMPLE POINT: This is a member of the sample space i.e. an outcome.

SIMPLE EVENTS: This relates to the outcome of a single trial.

COMPOUND EVENTS: This is a complex trial which result into many outcomes.

TYPES OF EVENTS AND THEIR PROBABILITIES

(A) CERTAIN OR SURE EVENTS: These are events that will occur irrespective of the circumstances. For instance, if today were to be Monday, it is certain that tomorrow will be Tuesday. That is a sure event. The probability of a sure event occurring is one (1).

(B) IMPOSSIBLE OR NULL EVENTS: These are events that can never occur. For example, if today were to be Monday, it is impossible that tomorrow will be Thursday. The probability of an impossible event occurring is zero (0).

(C) FAILURE OF COMPLEMENT OF AN EVENT: This is the non-occurrence of the event. So, if the probability of an event occuring is p, the complement of the event or the probability of the event not occuring is 1 - p. This is because the probability of an event occuring plus the probability of the event not occuring is 1.
If the probability that a student will pass an exam is 1/3; then the probability that the student will fail or not pass the exam is 1 - 1/3 = 2/3.
Similarly, if the probability of not winning a lottery is 3/5; the probability of winning the lottery will be 1 - 3/5 = 2/5. (D) MUTUALLY EXCLUSIVE EVENTS: Two events are said to be mutually exclusive if the occurrence of one automatically implies the non-occurrence of the other. For instance, if a coin is tossed, it either we have a tail (T) or a head (H). If we have a tail, it automatically means that we cannot have a head.  