User:Godfredabledu/Probability/Syllabus/Definitions of terminologies
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Definitions of terminologiesExperiment: Is a process that leads to one of several possible outcomes or Process which leads to well-defined results call outcomes. Some examples of experiments include simple process of checking whether a switch is turned on or off, counting the imperfections in a piece of cloth, tossing a die or a coin. Outcome: An outcome is the result of an experiment or other situation involving uncertainty. Drawing a card out of a deck of 52 playing cards is an experiment. One of the outcomes of the experiment may be that a queen of diamond is drawn. Sample Space: The set of all possible outcomes of a probability experiment is called a sample space and is usually denoted by S.
Sample Point: Each outcome in a sample space is an element or sample point. For example, when an experiment is performed, it gives rise to certain outcomes. These outcomes are called sample points( or elementary events). A collection of all possible outcomes or sample points is called sample space. A coin tossed once may either result in a head(H) or a tail(T). so there are only two outcomes of this experiment. Here, the sample space ,S, consists of only two sample points. Thus, S ={ H, T }. The number of sample points is 2. This is denoted n(S) = 2. A fair die thrown once may show up on its face either of the six numbers 1, 2, 3, 4, 5, and 6. There are six possible outcomes of this experiment and so the sample space is S ={1, 2, 3, 4, 5, 6 }.The number of sample points is n(S) = 6 Deterministic Experiment: An experiment is deterministic if its observed results are not subject to chance. In a deterministic experiment, if the experiment is repeated a number of times under exactly the same conditions, we expect the same results. For example, if we measure the distance between the points P and Q(in Km.) many times under the same conditions, we expect to have the same results. Random Experiment: An experiment is random if its outcomes are uncertain. If a random experiment is repeated under identical conditions, the outcomes may be different as there may be some random phenomena or chance mechanism at work affecting the outcomes. For example, tossing a coin or rolling a die is a random experiment since in each case the process can lead to more than one possible outcome. A synonym for random experiment is stochastic experiment. Trial: Each repetition of an experiment is a trial. For example, if a coin is tossed four times, each single toss is a trial. Equally Likely Outcomes: When any outcome of an experiment has the same chance of occurrence as any other outcome, then the outcomes are said to be equally likely. When a die is tossed once, the outcomes 1, 2, 3, 4, 5, and 6 are all equally likely as long as the die is fair. Event: An event is any collection of outcomes of an experiment. Formally, any subset of the sample space is an event. If we roll a die once, the event of rolling a “4” can be satisfied by only one outcome. That is the 4 itself. If we let A to represent an event, then A = { 4 }. The event of rolling an odd number can be satisfied by any one of three outcomes. These are 1, 3, and 5. If we let B represent an event, then B = {1, 3, 5 }. Any event which consists of a single outcome in the sample space is called an elementary or simple event. Events which consist of more than one outcome are called compound events. Mutually Exclusive Events: Two events, A and B, are said to be mutually exclusive if they cannot occur together. That is, two events, A and B, are mutually exclusive if the occurrence of A implies the non-occurrence of B and vice versa. Synonyms for mutually exclusive events are disjoint events, incompatible events, or non-overlapping events. For example if a die is tossed once, the number 4 and 5 cannot occur together and hence the event A = { 4 } and B={5}. are mutually exclusive. Note that for mutually exclusive events,A ∩ B; = ∅. Mutually Inclusive Events: Two events, A and B, are said to be mutually inclusive if they can occur together. Synonyms for mutually inclusive events are compatible events, or overlapping events. For example if a die is tossed once, the event A = { odd numbers } and B = { prime numbers} are mutually inclusive. Note that for mutually inclusive events, A ∩ B ≠ ∅. Collectively Exhaustive Events: Two or more events defined on the sample space are said to be collectively exhaustive it the union is equal to the sample space S. The events A ={odd numbers} and B = { even numbers } include every possible outcome in the sample space in the experiment of tossing a die once. Note that A = { 1, 3, 5 } and B = { 2, 4, 6 }. Thus A ∪ B ={1, 2, 3, 4, 5, 6 }. In an experiment if the events are collectively exhaustive, it means that at least one of the events must occur. If the set of events is collectively exhaustive and the events are mutually exclusive, the sum of the probabilities is equal to one. Note that in the experiment of tossing a coin once, S = { H,T } and the events A={H}.and B = { T }. The events are mutually exclusive because both A and b cannot occur at the same time. Also the events A and B are collectively exhaustive because A ∪ B ={H,T}. Thus, the sum of the probabilities of the events A and B is equal to 1. Independent Events: Two events A and B are said to be independent if the occurrence (or non-occurrence of one of them is not affected by the occurrence (or non-occurrence) of the other. For example, if two coins are tossed the event “Head” on the first coin and “Tail” on the second coin are independent. Dependent Events: Two events A and B are said to be dependent if the occurrence (or non-occurrence) of one of them is affected by the occurrence (or non-occurrence) of the other. For example, suppose a box contains 2 red pens and 3 blue pen and two are picked at random successively. The event “blue pen” in the second picking and the event “red pen” in the first picking are dependent
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