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Probability: Part 1 - Some Definitions with A Solved Problem Set- Continuous and Discrete Random Variables, Chebyshev Inequality


 
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Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE, Anyone else who needs this Tutorial as a reference!

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Probability - Part 1 - Basic Probability Definitions, Random Variables

Here's a quick outline of the topics covered in this tutorial :

1. A Review of Sets

    Definitions of the Universal Set, Null Set,  Union and Intersection of Sets, Mutually Exclusive Events, Exhaustive Events

2. Basic Probability Definitions - Experiment, Random Experiment, Sample Space, Baye's Theorem, Independent Events:


• Experiment :
Any happening under observation.
• Random Experiment :
An experiment whose outcome can not bepredicted.
• Sample Space: 
The set of all possible outcomes of a random sample.
• Event :
It is the subset of sample space.
• Classical Definition of probability :
Suppose a random sample has N possible outcomes, exhaustive and equally likely. Let M of these be favourable to the happening event A, then
P = M/N
• Relative Definition of probability
• Bayes Theorem
• Independent Events

3. Dealing with Random Variables:


Random Variable :

It is simply a value of a measurement which is associated with our experiment, i.e. when random variable X takes value x, its probability P(X=x) is defined.
Random variable could be of 2 types :

(i) Discrete Random Variable :


When X takes finite number of values.

(ii) Continuous Random Variable :


When X takes all possible values in an interval.

Discrete Random Variable: 

Probability Mass Function, Cumulative Distributive Function, Expectation and Variance, Moments, Moment Generating Functions (Mgf)

Continuous Random Variable: 

Probability Density function (pdf),  Cumulative Distribution function (cdf), Expectation and Variance, Moments, Moment Generating Functions (Mgf)

Chebyshev’s Inequality :


We know that σ2 gives a measure of deviation from the mean. Chebyshev’s inequality gives us a way of understanding how σ2 measures the probability of these deviations. (Refer to standard textbooks for proof.)

Here are some of the problems solved in this tutorial :


Q: Let Band C be disjoint events. If A and B are independent, and A

and C are independent, prove that A and B C are independent.
Q: In a certain colony, 60% of the families own a car, 30% own a house and 20% own both a car and a house. If a family is randomly chosen, what is the probability that this family owns a car or a house but not both?
Q: If four married couples are arranged to be seated in a row, what is the probability that no husband is seated next to his wife?
Q: The entries in a 2x2 matrix are integers that are chosen randomly and independently, and for each entry, the probability that the entry is odd is p. If the probability that the value of its determinant is even is ½, then find the value of p2.
Q: A gambler tosses an unbiased normal coin unless he is either ruined or the coin has been tossed for a maximum of five times. For each head he wins a rupee and for each tail, he looses a rupee. Find the probability that the gambler is ruined.
Q: Football clubs A and B are set to play a series of three games against each other to decide the league champion. The probabilities of A winning, losing and drawing a game are (1/2), (3/8), (1/8) respectively. A club gets 3 points for winning, 1 for a draw and 0 for losing. What is the probability that (i) A will win the league (ii) B will win the league (iii) the league will end in a tie?
Q: Let four fair die be rolled at once. The outcomes on the die be denoted by a, b, c, d respectively. Let E denoted the event that absolute value of (a – 1)(b – 2)(c – 3)(d – 6) = 1. Find P(E).
Q: In any given year a male a male policy holder will make a claim with probability pm and a female will make a claim with probability pf. Fraction of mails holding the policy = α. Policy is maker chosen randomly and Ai denotes probability that this policy holder will make a claim in the year (i = 1,2). Find Pr(Ai), Pr(A2|A1).
Q: A bag contains 10 balls of which some are black and the others are white. If three balls are being drawn w/o replacement and all of them are found to be black, find the probability that the bag contains 1 white and 9 black balls.
Q: There are two packs A and B of 52 playing cards. All the four aces from the pack A are removed whereas from the pack B, one ace, one king, one queen and one jack is removed. One of these two packs is selected randomly and two cards are drawn simultaneously from it, and found to be a pair. Find the probability that pack A was selected.
Q: In a production line ICs are packed in vials of 5 and sent for inspection. The probabilities that the number of defectives in a vial is 0,1,2,3 are 1/3, 1/4, 1/4, 1/6 respectively. Two ICs are drawn at random from a vial and found to be good. What is the probability that all ICs in this vial are good?
Q: A fair die is thrown 3 times. What is the probability that sum of three numbers appearing on the die is less than 11?
Q: The number of customers that visit a shop is a random variable with mean 18 and standard deviation 2.5 with what probability can we assert that there will be between 8 to 28 customers?


Complete Tutorial with Solved Problems and Solutions :