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The set of all possible outcomes of a random experiment is called the sample space (S). Each possible outcome of a random experiment is called an elementary outcome(E).
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If A and B are any two events then they are said to be mutually exclusive if AnB = Ø. Namely, the intersection set of events must be empty set if they are mutually exclusive.
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Having a "head" and having a "tail" when throwing a coin are mutually exclusive.
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In the classical probability approach to assign a probability to an event, the assumption is that all the outcomes have the same chance of happening.
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The empirical probability is based on experperiments.
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Sometimes it may not possible to observe the outcomes of events; therefore, the researcher may assign a probability to an event. In subjective probability approach, the researcher assigns a suitable value as the probability of the event. Therefore, a personal judgement comes in to play to assign the probability.
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When counting the possible outcomes of an event, it is sometimes important to distinguish between ordered and unordered arrangements. When order is important, we call the arrangements of a finite number of distinct objects a permutation. When order is not important, the arrangements of r objects from n distinct objects is called a combination.
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Since the order of students is not important we can use combination. Thus: C(10,3)=10!/7! 3!=7!*8*9*10/7!*3*2=4*3*10=120. So 120 different groups are possible.
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In 10 days either 2 projects will finish(P(2)) or 1 project will finish(P(1)) or none (zero) project will finish (P(0)) in 10 days. Thus: P(0)+P(1)+P(2)=1 We are interested in at least one project to finish. So we have to compute P(1)+P(2) P(1)+P(2)=1-P(0) Since P(Ali Finish)=0.4, then P(Ali Doesnt Finish)=1-0.4=0.6 Since P(Aysel Finish)=0.7, then P(Ali Doesnt Finish)=1-0.7=0.3 Thus P(0)=P(Neither Ali nor Aysel Finish)=06*0.3=0.18 Therefore: P(1)+P(2)=1-P(0)=1-0.18=0.82
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Suppose that we know that event B has occurred and we are interested in finding the probability of event A. That is, we are interested in finding the probability of A knowing that event B has occurred. This is denoted by P(A|B) and is defined as: P(A|B=P(AnB)/P(B)
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If the dice is greater than 3, then the event space is S=(4,5,6) So, if we want it to be smaller than 6, we want the dice come 4 or 5. Therefore the probability is 2/3.
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The probability of not being pink=13/20 The probability of being blue=5/20 Then probability of being blue, given not being pink=(5/20)/(13/20)=5/13
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Let P(A) denote having a head and P(B) having a number smaller than 3. P(A)=1/2 P(B)=1/3 (since we want either 1 or 2 come on dice; 2/6=1/3) Since these are independent events P(AnB)=P(A)*P(B)=1/2*1/3=1/6
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Binomial distribution is a widely employed discrete probability distribution in statistics where a set of independent observations constitutes exactly two disjoint outcomes of a trial. Therefore in binomial distribution, an outcome of a random experiment can be classified under two different categories. For example, when a die is tossed once we observe six different numbers, that is x = 1, 2, 3, 4, 5 and 6. If we classified these outcomes as even or odd numbers then we will get two different outcomes.
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A random experiment (trial) with only two possible outcomes is frequently used and called a Bernoulli trial with the following properties: 1. Each trial has only two possible outcomes, such as head and tail, 0 and 1 or success and failure. 2. The trials are independent experiments. 3. The probability of success, denoted by and the probability of failure, denoted by remains constant for all trials (p + q = 1).
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The Poisson distribution is widely used for discrete probability distribution which is used to model the number of outcomes occurring during a specified time interval or in a definite region. Here, the time interval indicates any length, for instance an hour, a day, or a month. Also, the definite region term means that a piece of line segment, an acre, size of a football field, a volume.
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In binomial distribution the sampling process is carried out with replacement. On the other hand hypergeometric distribution doesn’t involve with independence assumption for each trial and accordingly the sampling process is established on without replacement. Because of these features, hypergeometric distribution is broadly used in various real life applications especially for acceptance sampling in quality control
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A major difference between continuous and discrete random variables is the former takes on uncountable and infinite number of possible outcomes in a given interval. Hence the range of continuous random variable X comprises all real numbers in an interval. In addition to the above given illustrations, water consumption amount in a household, weights of people in a population, the speed of wind in a open certain area, waiting time in a supermarket, checkout lanes or load on a bridge are the few examples for continuous random variable for real world applications. From these examples it’s clear that random variable X can take unaccountably infinite values.
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The probability density function f (x) of the uniform continuous random variable X takes a constant value over the range of the random variable X is defined.
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the minimum value of x is a=20 the maximum value of x is b=40 Mean=0.5(a+b)=30 Variance=(b-a)(b-a)/12=100/12=25/3=8.3333...
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The simpliest solution is geometric. P(x<38) is equal to the area of the rectangle whose short size is 0.05 (1/(40-20)=0.05) and long size is 18 (38-20=18). Thus P(x<38)=0.05*18=0.9
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Normal distribution is one of the most significant and extensively used continuous probability distribution. The major reason for this circumstance is majority of the continuous random variables which are observed through real life applications (social, medical, physical, biological) are normally or approximately normally distributed (bell-shaped) variables. Furthermore, normal distribution provides basis for the statistical inference. Because of its shape, normal distribution occasionally called “bell curve” or also called “Gaussian curve” which was developed by a mathematician Karl Friedrich Gauss. In addition to common properties of distribution functions: Normal distribution is symetric around the mean. The probability density function f (x) does not the touch and intersect x axis.
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Exponential distribution is another most significant and extensively used continuous probability distribution. Exponential random variable is frequently used to model the time interval between two events. Some illustrations of random variables that generally conform to model by means of exponential distribution are presented below. • Arrival time between two customers. • Time between two messages. • Time between telephone calls received by a customer service. • Time between customers who are arriving to the checkout lane of the supermarket. • Time between two failures of a certain mechanical device.
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