Introductory Probability- Compound and Independent Events, Mutually Exclusive Events, Multi-Stage Experiments

• On the most basic level, probability is defined as : P(x)=number of favourable outcomes/total number of outcomes
• Here, P(x) is the probability of occurrence of an event x, and the event x here is a discrete and independent event.
• The definition of probability in general can be for continuous events, dependent events, conditional events and so on.
• In probability theory, we perform or consider the experiments in which all possible outcomes are known in advance.
• P(x)+P(x')=1 ;that is, sum of probability of an event occurring and the probability of the same event not occurring is one.
• Compound Events are formed by combining two or more simple events:
• A∩B:
• A∪B :
• P(A∪B)=P(A)+P(B)-P(A∩B)
• Two events A and B are said to be Mutually Exclusive Events if P(A∩B)=0, that is, if both A and B can NEVER happen simultaneously.
• Two events are said to be Independent, if P(A|B)=P(A), or the conditional probability of event A occurring given event B has occurred is equal to the probability of event A, or that event A is unaffected by event B
• Conditional Probability: P(A|B)=P(A∩B)P(B)
• If A and B are independent then, P(A∩B)=P(A)P(B)
• An experiment can be single stage or multi-stage, depending upon if an experiment can be further broken down into stages.
• In case of a multi-stage experiment, we can have conditional probabilities between events A and B such that either :
• A and B are in the same stage. In this case we use: P(AB)=n(A∩B)n(B)
• A is in the stage that has occurred, and B is in a stage that will occur (experiment is not yet over).
• A is in a previous stage, and B is in a later stage, and the experiment is over.
• In the second and third case, we make use of Baye’s networkBaye’s theorem

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Application of Probability and a quick example of a Probability Distribution, Random Variables and Sample Space

• Probability finds its application in a wide variety of areas. Be it Artificial Intelligence, communication, statistical analysis, quantum mechanics and many more.
• One way of using probability is introducing Random variables. If S is the sample space, and w is an element in sample space, then a real valued function X which assigns a unique real number X(w) to each w in S is called a random variable X.
• Example: Let a coin be tossed twice. S={HH,HT,TH,TT}
  
  
  
• There can be a probability distribution associated with a random variable. If a random variable X takes values x1,x2,x3…,xn with probabilities p1, p2, p3, …, pn, then probability distribution is given by the following table. Each pi>=0 and Sum of all p
  

  X	x1	x2	x3	…	…	xn
  P(X)	p1	p2	p3	…	…	pn

Here are some of the problems covered in this tutorial :

Q: There are three events A, B and C one of which must, and only one can happen, the odd are 8 to 3 against and 2 to 5 for B. Find the odd, against C.