This document discusses key concepts in probability. It defines basic terms like experiment, sample space, event, and probability. It provides examples of calculating probability for coin tosses and dice rolls using the classical method of dividing the number of ways an event can occur by the total number of possible outcomes. The document also discusses limitations of the classical method and introduces the empirical and subjective methods of determining probability based on observed frequencies and personal judgment respectively.

1. Probability
PRESENTED BY-
VARUN SAXENA
MBA (GEN.)

2. Basic Terms of Probability
 In probability, an experiment is any process that can be repeated in which the
results are uncertain.
 A simple event is any single outcome from a probability experiment.
 Sample space is a list of all possible outcomes of a probability experiment.
 An event is any collection of outcomes from a probability experiment.

3. Example
 Experiment : Tossing a coin
 Sample Space: { Head, Tail)
 Event: (Only Head wants) : {Head}

4. Probability
 The probability of an event, denoted P(E), is the likelihood of that event occurring.
 The Probability of an event :
P(Event) =
𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑖𝑡 𝑐𝑎𝑛 ℎ𝑎𝑝𝑝𝑒𝑛
𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

5. Example
 When a coin is tossed, there are two possible outcomes: Heads and Tails
P(Heads) = ½
 When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6.
P(1) = 1/6.

6. Properties of Probability
 The probability of any event E, P(E), must be between 0 and 1 inclusive. That is, 0
< P(E) < 1.
 If an event is impossible, the probability of the event is 0.
 If an event is a certainty, the probability of the event is 1.
 If S = {e1, e2, …, en}, then P(e1) + P(e2) + … + P(en) = 1.

7. Three methods for determining
Probability
 Classical method
 Empirical method
 Subjective method

8. Classical Method
 The classical method of computing probabilities requires equally likely outcomes.
 If an experiment has n equally likely simple events and if the number of ways that
an event E can occur is m, then the probability of E, P(E), is

9. Example for classical method
 Let us suppose a bag of balls contains 6 brown balls, 15 yellow balls, 5 red balls,
20 orange balls, 11 blue balls, and 4 green balls. Suppose that a ball is randomly
selected.
 What is the probability that it is brown?
 P(Brown) = 6/61

10. Limitations of Classical method
 Classical method fails if the number of outcomes of the random experiment is
infinite.
 If the various outcomes of random experiment are not equally likely.
 If the actual number of possible outcomes is not known.

11. Empirical method
 The probability of an event E is approximately the number of times event E is
observed divided by the number of repetitions of the experiment.
 The empirical probability, also known as relative frequency.
 The empirical approach to probability is based on law of large numbers.
 So to achieve more accuracy in the result, collect more observations which provide
more accurate estimate of the probability.

12. Example for Empirical method
 A coin is thrown 100 times out of which head appears 12 times. Find the
experimental probability of getting the head?
The coin is thrown 100 times. So total number of trails =100
Given head occurs 12 times. So the number of times the required event occurs =
12
Therefore probability of getting the event of head =
12
100
= 0.12

13. Subjective Probability
 Subjective probabilities are probabilities obtained based upon an educated guess.
 A subjective probability describes an individual's personal judgment about how
likely a particular event is to occur.
 A person's subjective probability of an event describes his/her degree of belief in
the event.
 For example, there is a 10% chance of rain tomorrow.

14. Thanks