Probability.pptx

PROBABILITY Probability Probability denotes the possibility of the outcome of any random event. The meaning of this term is to check the extent to which any event is likely to happen. For example, when we flip a coin in the air, what is the possibility of getting a head? The answer to this question...

PROBABILITY Probability Probability denotes the possibility of the outcome of any random event. The meaning of this term is to check the extent to which any event is likely to happen. For example, when we flip a coin in the air, what is the possibility of getting a head? The answer to this question is based on the number of possible outcomes. Here the possibility is either head or tail will be the outcome. So, the probability of a head to come as a result is 1/2. The probability is the measure of the likelihood of an event to happen. It measures the certainty of the event. The formula for probability is given by; P(E) = Number of Favorable Outcomes/Number of total outcomes P(E) = n(E)/n(S) Here, n(E) = Number of event favorable to event E n(S) = Total number of outcomes Probability is simply a way of measuring how likely some event is to happen. Common ways of reporting probability are: - fractions - decimals - percentages Converting between these different values uses simple mathematical functions: fraction to decimal - divide the values in the fraction (ex: 1/2 =.5) decimal to percentage - multiply by 100 (ex:.5 * 100 = 50%) number of successful or desired outcomes ------------------------------------------------------------ total number of possible outcomes Basic concepts The field of probability theory deals with the analysis of random phenomena. There are two different types of random phenomena: events and experiments. Experiments produce a list of outcomes, such as flipping a coin repeatedly to see what each side is or throwing a dice to generate numbers between 1 and 6. Mathematically, experiments are usually described as the outcome of a single trial. The probability of an event is solely dependent on the probability of the experiment from which it is derived and not from other events. For example, if you flip a coin 20 times and get 15 heads, you will get 15 heads with 95% confidence as each coin flip is independent of any other one. Events are also described as likelihood of any event e.g. probability of rain. The probability of an event is how likely it is that the event will occur. Probability questions exist everywhere and at all times. Most importantly, probability lets us make predictions even though we cannot see into the unknowable future. Imagine a coin that you toss. There are two outcomes when throwing a coin: it lands on heads or tails. But there is an infinite number of possible outcomes —you can throw the coin many times and get a different answer every time. And yet we still talk about “heads” and “tails. Mathematicians define the probability of the occurrence of an event on a set of possible outcomes as “the ratio of the number of ways in which it can happen to all possible outcomes.” The basic probability concept is defined as the measure of the chance that an event will occur. A numerical value indicates the possibility of an event happening out of many possible outcomes. The values taken by probability are usually between 0 and 1, where 0 or 1 There are various terms utilized in the probability Experiment A trial or an operation conducted to produce an outcome is called an experiment. Random Experiment An experiment whose result cannot be predicted, until it is noticed is called a random experiment. For example, when we throw a dice randomly, the result is uncertain to us. We can get any output between 1 to 6. Hence, this experiment is random. Sample Space A sample space is the set of all possible results or outcomes of a random experiment. Suppose, if we have thrown a dice, randomly, then the sample space for this experiment will be all possible outcomes of throwing a dice, such as; Sample Space = { 1,2,3,4,5,6} Trial A trial denotes doing a random experiment. Favorable Outcome An event that has produced the desired result or expected event is called a favorable outcome. For example, when we roll two dice, the possible/favorable outcomes of getting the sum of numbers on the two dice as 4 are (1,3), (2,2), and (3,1). Random Variables The variables which denote the possible outcomes of a random experiment are called random variables. They are of two types: Discrete Random Variables Discrete random variables take only those distinct values which are countable. Continuous Random Variables Whereas continuous random variables could take an infinite number of possible values. Independent Event The probability of one event does not change based on the outcome of the other event. When the probability of occurrence of one event has no impact on the probability of another event, then both the events are termed as independent of each other. For example, if you flip a coin and at the same time you throw a dice, the probability of getting a ‘head’ is independent of the Mutually Exclusive The events are mutually exclusive if they do not occur simultaneously. This indicates that two events cannot happen at the same time, the occurrence of one event results in non occurrence of other event. For example, consider the following two events: A) rolling a ‘2’ and B) rolling an odd number. Since 2 is an even number, it's not possible to roll a 2 and for that number to be odd. Therefore, these events are mutually exclusive. Exhaustive Events Exhaustive events are a set of events in a sample space such that one of them compulsorily occurs while performing the experiment. In simple words, we can say that all the possible events in a sample space of an experiment constitute Laws of Probability 1. The probability that two events will both occur can never be greater than the probability that each will occur individually. 2. If two possible events, A and B, are independent, then the probability that both A and B will occur is equal to the product of their individual probabilities. Example: Consider one more example. Suppose a bag contains 5 white and 3 red marbles. Two marbles are drawn from the bag one after another. Consider the events A = Drawing a white marble in the first draw. B = Drawing a red marble in the second draw. If the marble drawn in the first draw is replaced back in the bag, then A and B are independent events because P(B) remains the same whether we get a white marble or a red marble in the first draw. 3. If an event can have a number of different and distinct possible outcomes, A, B, C, and so on, then the probability that either A or B will occur is equal to the sum of the individual probabilities of A and B, and the sum of the probabilities of all the possible outcomes (A, B, C, and so on) is 1 (that is, 100 percent).