Quantitative Aptitude > Probability

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  • A probability experiment, also called a random experiment, is an experiment for which the result, or outcome, is uncertain. We assume that all of the possible outcomes of an experiment are known before the experiment is performed, but which outcome will actually occur is unknown. The set of all possible outcomes of a random experiment is called the sample space, and any particular set of outcomes is called an event.

    For e.g. consider a cube with faces numbered 1 to 6, called a 6-sided die. Rolling the die once is an experiment in which there are 6 possible outcomes— either 1, 2, 3, 4, 5, or 6 will appear on the top face. The sample space for this experiment is the set of numbers 1, 2, 3, 4, 5, and 6. Two examples of events for this experiment are (i) rolling the number 4, which has only one outcome, and (ii) rolling an odd number, which has three outcomes.

  • The probability of an event is a number from 0 to 1, inclusive, that indicates the likelihood that the event occurs when the experiment is performed. The greater the number, the more likely the event.

    Ex.  Consider the following experiment. A box contains 15 pieces of paper, each of which has the name of one of the 15 students in a class consisting of 7 male and 8 female students, all with different names. The instructor will shake the box for a while and then, without looking, choose a piece of paper at random and read the name. Here the sample space is the set of 15 names. The assumption of random selection means that each of the names is equally likely to be selected. If this assumption is made, then the probability that any one particular name is selected is equal to . For any event  the probability that occurs, denoted by , is defined by the ratio

    If M is the event that the student selected is male, then

  • In general, for a random experiment with a finite number of possible outcomes, if each outcome is equally likely to occur, then the probability that an event  occurs is defined by the ratio

    In the case of rolling a -sided die, if the die is "fair," then the  outcomes are equally likely. So the probability of rolling a  is  , and the probability of rolling an odd number—rolling a 1, 3, or 5—can be calculated as

  • The following are general facts about probability.
    • If an event is certain to occur, then .
    • If an event  is certain not to occur, then .
    • If an event  is possible but not certain to occur, then .
    • The probability that an event will not occur is equal to .
    • If  is an event, then the probability of  is the sum of the probabilities of the
    • outcomes in .
    • The sum of the probabilities of all possible outcomes o
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