Some terminologies Experimental Probability - is the other type of probability, which is based on experiments frequency of an outcome/number of trials. - it is the ratio of the number of times an outcome occured to the total of events or trials. experimental probability = frequency of an outcome/total number of trials. Theoretical Probability - mathematically determined. Simulation - is the process of finding the experimental probability Empirical Study - performing experiment repeatedly, collect, and combine the data and analyzing the results. Probability Histogram - a bar chart used for data involving probabilities
1. CountingArrangements/Outcomes one way to count the number of possible outcomes graphically is tree diagram. the list of all possible outcomes is called the sample space. event - a collection of one or more outcomes in the sample space
COUNTING ARRANGEMENTS = HOW MANY ARRANGEMENTS? USE FACTORIALFACTORIAL - is the product of all positive integers from n counting backwards to 1, denoted by n!.Example: find 5!5! means 5 x 4 x 3 x 2 x 15! = 5 x 4 x 3 x 2 x 15! = 120SAMPLE PROBLEM:How many ways can you arrange your 10 books in a shelf?Answer: USE FACTORIAL 10! = 3,628,800
2. THE FUNDAMENTAL PRINCIPLE OF COUNTINGFundamental Principle of Counting says that: “If event M can occur in m ways and is followed by an event N that can occur in n ways, then the event M followed by event N can occur in n times m ways.”EXAMPLE, A and B are two events. To know how many ways A and B can occur, multiply the number of ways for A by the ways for B.SAMPLE PROBLEM: If a restaurant offers 10 different types of burgers, 5 different types of pizza, and 3 different beverages, how many combinations/meals can you pick?ANSWER: use FPC!Let x = number of mealsM = 10 x 5 x 3M = 150Then, you can choose from 150 meals.
3. PERMUTATION ANDCOMBINATIONPERMUTATION - arrangement or listing of numbers in which the order does matterFORMULA: nPr = n! / (n-r)!where n = total number of possible outcomes r = number of items taken at a timeCOMBINATION - arrangement or listing of numbers in which the order does not matterFORMULA: nCr = n! / (n-r)!r!where n = total number of possible outcomes r = number of items taken at a time
4. some types of events1) simple event - a single event2) compound event - two or more simple events3) independent event - two or more events wherein one event do not affect other events outcome4) dependent event - two or more events wherein one event affect the others outcome5) mutually exclusive event - two events wherein one event cannot happen with the second event at the same time6) inclusive events - two events in which one event can happen at the same time with the
5. finding the probabilities. THE PROBABILITY OF INDEPENDENT EVENT - if two events are independent, then the probability of both A and B to occur is the product of the individual probabilities of the two events. P(A and B) = P(A) times P(B) THE PROBABILITY OF DEPENDENT EVENT - if two events are dependent, then the probability of both A and B to occur is the product of the probability of A and the probability of B after the event A. P(A and B) [dependent] = P(A) times P(B after A)
THE PROBABILITY OF MUTUALLY EXCLUSIVE EVENTS - if two events are mutually exclusive events, then the probability that both A or B to occur is the sum of the probabilities of A and B. P(A or B) = P(A) + P(B) THE PROBABILITY OF INCLUSIVE EVENTS - if two events are inclusive events, then the probability for both A or B to occur is the sum of the probabilities of A and B decreased by the probability of both A and B to occur. P(A or B) = P(A) + P(B) - P(A and B)