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- 1. PROBABILIT Y PRESENTED BY:ISH,DIVESH & GAURAV COURSE:B.SC(H) MATHS
- 2. What is probability? Probability is the measure of how likely an event or outcome is. Different events have different probabilities!
- 3. How do we describe probability? You can describe the probability of an event with the following terms: certain (the event is definitely going to happen) likely (the event will probably happen, but not definitely) unlikely (the event will probably not happen, but it might) impossible (the event is definitely not going to happen)
- 4. How do we express probabilities? Usually, we express probabilities as fractions. The numerator shows the POSSIBLE number of ways an event can occur. The denominator is the TOTAL number of possible events that could occur. Let’s look at an example!
- 5. What is the probability that I will choose a red marble? In this bag of marbles, there are: 3 red marbles 2 white marbles 1 purple marble 4 green marbles
- 6. Ask yourself the following questions: 1. How many red marbles are in the bag? 2. How many marbles are in the bag IN ALL? 3 10
- 7. This is denoted with ‘S’ and is a set whose elements are all the possibilities that can occur A probability model has two components: A SAMPLE SPACE and an assignment of probabilities related with each OUTCOME. ***Each element of S is called an outcome. A probability of an outcome is a number and has two properties: 1. The probability assigned to each outcome is nonnegative. 2. The sum of all the probabilities equals 1.
- 8. Let's roll a die once. S = {1, 2, 3, 4, 5, 6} This is the sample space---all the possible outcomes ( ) Number of ways that can occur Number of possibilities E P E = probability an event will occur What is the probability you will roll an even number? There are 3 ways to get an even number, rolling a 2, 4 or 6 There are 6 different numbers on the die. ( ) 3 1 Even number 6 2 P = =
- 9. The word and in probability means the intersection of two events. What is the probability that you roll an even number and a number greater than 3? E = rolling an even number F = rolling a number greater than 3 ( )P E F∩ How can E occur? {2, 4, 6} How can F occur? {4, 5, 6} {2,4,6} {4,5,6} {4,6}E F∩ = ∩ = 2 1 6 3 = = The word or in probability means the union of two events. What is the probability that you roll an even number or a number greater than 3? ( )P E F∪ 4 2 6 3 = = {2,4,6} {4,5,6} {2,4,5,6}E F∪ = ∪ =
- 10. Independent Events10 Two events are independent if The outcome of event A, has no effect on the outcome of event B. Example: "It rained on Tuesday" and "My chair broke at work“ are not at all related to each other. When calculating the probabilities for independent events you multiply the probabilities.
- 11. LETS TAKE AN EXAMPLE: Let Event A:Today it will rain in Delhi. Event B:Today I will reach office late. Event C:Possibility of both events A&B happening together. GIVEN:P(A)=0.1,P(B)=0.02 Hence ; P(C)=P(A)*P(B) =0.1*0.02 =0.002
- 12. •Conditional Probability The conditional probability of an event A (given B) is the probability that an event A will occur given that another event, B, has occurred.
- 13. FORMULA S A B ( ) ( ) ( ) P A B P A B P B ∩ =
- 14. EXAMPLE Suppose you roll a pair of dice: one RED in colour while other is GREEN. The probability that the sum of the numbers on the dice = 9 is 4/36 since there are 4 of the 36 outcomes where the sum is 9: (3,6) (4,5) (5,4) & (6,3). What if you see that the RED die shows the number 5, but you still haven’t seen the green die? What are the chances then that the sum of numbers we get after rolling pair of dice is 9 ??????
- 15. Let B = event “5 on Red die” when pair of dice is rolled. Let A=event “sum is 9”. Using the formula below;we get the answer as 1/6.TRY IT OUT !!!!!!!!!!!!! ( ) ( ) ( ) P A B P A B P B ∩ =
- 16. Complementary Events 16 One event is the complement of another event if the two events do not contain any of the same outcomes, and together they cover the entire sample space.
- 17. E This is read as "E complement" and is the set of all elements in the sample space that are not in E Remembering our second property of probability, "The sum of all the probabilities equals 1" we can determine that: ( ) ( ) 1P E P E+ = This is more often used in the form ( ) ( )1P E P E= − If we know the probability of rain is 20% or 0.2 then the probability of the complement (no rain) is 1 - 0.2 = 0.8 or 80%
- 18. GAMBLER’S FALLACY
- 19. The gambler's fallacy, also known as the Monte Carlo fallacy , is the mistaken belief that if something happens more frequently than normal during some period, then it will happen less frequently in the future (presumably as a means of balancing nature).For eg : A coin is tossed 5 times, and all the results comes as ‘ HEADS ’. So during the 6th attempt , the person may feel that A TAIL IS DUE and hence falls prey to the Fallacy by predicting a Tail ,not thinking that the PROBABILITY OF OBTAINING EITHER HEADS OR TAILS REMAINS SAME The use of the term Monte Carlo fallacy originates from the most famous example of this phenomenon, which occurred in a Monte Carlo Casino on 18/8/1913.Then,a ball fell in black 15 TIMES . Because of such an occuring, gambler’s started betting on RED.But they fell prey to the Fallacy as RED TURNED-UP after 26 attempts.
- 20. THANK YOU FOR YOUR ATTENTION 20

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