Solved Problems – Baye’s Theorem

Q1. There are three bags. First bag contains 1 white, 2 red and 3 green balls. Second bag contains 2 white, 3 red and 1 green balls. Third bag contains 3 white, 1 red and 2 green bals. A bag is chosen at random and 2 balls are drawn from it. These are found to be 1 white and 1 red. Find the probability that the balls so drawn came from the second bag. Solution: Let A1 be the event of choosing first bag Let A2 be the event of choosing second bag…continue reading →

Baye’s Theorem

Bayes theorem - Also known as Theorem of Inverse Probability Let A1,A2,...An be a set of exhaustive and mutually exclusive events of the sample space S with for each i. Let E be any event in S. Then Proof We have Therefore, Therefore,continue reading →

Solved Problems-III:Probability Theory

Q1: There are 3 groups of children, containing 3 girls and 1boy, 2 girls and 2 boys, 1 girl and 3 boys respectively. One child is selected at random from each group. Find the probability of selecting 1 girl and 2 boys. Solution: Let G be the event of selecting girl Let B be the event of selecting boy Different cases and associated probabilities are given below: Case1: Group1: G, Group2: B, Group3: B - Probability = ¾ x 2/4 x ¾ Case 2: Group1: B, Group2: G, Group3: B - Probability =…continue reading →

Solved Problems-II:Probability Theory

Q1: A box of chocolates contain 3 orange, 5 green and 6 red chocolates. If one is taken at random, what is the probability that it is either orange or red? Solution: Total number of chocolates = 14 Favourable cases = 9 Hence answer = 9/14 Q2: A box of chocolates contain 4 orange, 5 green and 6 red chocolates. If 2 chocolates are taken at random, what is the probability that : Both are red One is green and one is orange Solution: Total number of chocolates = 15 Number of ways…continue reading →

Solved Problems-I: Probability Theory

Q1: Find the probability that a leap year selected at random will contain 53 Sundays. solution : A leap year contains 366 days which include 52 complete weeks and two additional days. These two days can be: (1). Monday and Tuesday ,(2). Tuesday and Wednesday, ... and so on. Of these 7 possibilities only 2 contain a Sunday. Hence probability that a leap year has 53 Sundays = 2 / 7 Q2: Two dice are rolled. Find the probability that both the dice show the same number Solution: 6 / 36 = 1…continue reading →

Properties and Theorems on Probability

Properties (1) ; where denotes the null set. We have By probability axiom, Therefore, (2) If A is an event and is its complement, then, We have = S Therefore,   Note: If P(E) = 1, then E is called a sure event and if P(E) = 0, then E is called an impossible event. Addition theorem on 2 events If A and B are any two events, then Note: If A and B are mutually exclusive, Then So, Addition theorem for 3 events If A,B and C are any three events, Note:…continue reading →

Defining Probability

There are three approaches to defining probability Mathematical or Classical Empirical Axiomatic Mathematical or Classical Definition of probability If there are n exhaustive mutually exclusive and equally likely cases, of which m cases are favourable to the happening of an event A, then the probability of A is defined as   Clearly, Examples: Probability of getting H in a coin toss experiment = Probability of getting 6 in a die rolling experiment = Probability of drawing a spade from a deck of 52 cards = Empirical Definition of Probability The mathematical definition of…continue reading →
Events

Events

As discussed in the previous posts, a random experiment will have multiple possible outcomes. While conducting an experiment, we may be interested not just in one outcome, but in a set of outcomes. For example, in the experiment of rolling a die, the event of getting an odd numbered face represents a set of possible outcomes :- set   Similarly, the event of getting an even numbered face is another set of possible outcomes :- set It must be noted that, the above defined sets  and   are subsets of the sample space…continue reading →
Introducing Probability

Introducing Probability

The term “probability”, is used in our everyday life, to denote the degree of belief in a particular outcome of an experiment. Some examples: The chance for a candidate to win the election The probability of getting stuck in a traffic jam during the morning hours The chance for a defective part to come out of the assembly line, in a batch of 1000 Random Experiment An experiment or trial having more than one possible outcome is referred to as a random experiment. Outcomes of such experiments cannot be predicted with certainty. In the…continue reading →