Probability. Definitions Law of large numbers Addition rule Multiplication rule Permutation and combination

Transcription

1 Probability Definitions Law of large numbers Addition rule Multiplication rule Permutation and combination

2 Definitions Event: an outcome of a random process Simple event: an event that may not be simplified any further or can happen in only one way Sample space: Collection of all the simple events

3 Definition Relative frequency Classical approach

4 Law of Large Numbers If a procedure involving a random process repeated for a large number of times then the relative frequency approaches the true/actual probability

5 Probability P(A) = 0 P(A)=1 0<=P(A)<=1 Sum of all the P(A) is 1

6 Probability of Complement Note that P(A) + P(A-bar) = 1 P(at least one) = 1 P(None)

7 Odds Actual odds against an event A Actual odds in favor of an event Payoff odds = Ratio of net profit to amount bet Examples: P(passing the class) = 0.80 P(IRS audit) = 0.01 P(winning a scratcher) = 0.05

8 Mutually Exclusive Events Mutually exclusive events: Events that may not happen simultaneously Examples: People younger than 21 vs. older than 21 Being on time vs. not Being an American vs. not Their intersection is empty set

9 Addition Rule P(A or B) = P(A) + P(B) P(A and B)

10 Addition Rule Examples: iphone and ipad Visa and MasterCard P(passing the class) = 0.80 Target vs Walmart

11 Independent Events Independent events: Occurrence of one event does not change the probability of occurrence of the other Examples: Occurrence of lung cancer and smoking Age information and texting while driving Sexual behavior and age group 5% Rule for independence

12 Multiplication Rule P(A and B) = P(A). P(B A) Examples: Quality Control: Garments manufacturing in Bangladesh Probability of being born on the same day of the week, same month of the year, any day of the year Quiz with one multiple choice and one T/F

13 Fundamental Rule for Counting M x N Examples: ATM cards Executive positions at a business among three potential candidates Factorial!

14 Permutation and Combination Permutation: Selection of x different items from a group of n different items without replacement where order does matter Examples: President, Vice President, Treasurer

15 Combination Selection of x different items from a group of n different items without replacement where order DOES NOT matter Age Discrimination example at work Mortgage example

16 Random Variables A variable that takes on values that are the outcome of a random process Exact outcome unknown until the event happens Examples: Tossing a fair coin or rolling a die The time to experience a mechanical failure for an automobile or a plane The outcome of a sports event such as WC Final The gender of a baby that is yet to be conceived

17 Two Types of Random Variables Discrete random variables Examples: The outcome of game or rolling a die Number of people attends a local fair Continuous random variables Time to experience a mechanical failure for an automobile or plane

18 Probability Distribution Probability associated with each value of the random variable is between 0 and 1 All the probabilities that are associated with random variable x add up to 1 Examples: Tossing a fair coin Rolling a die

19 Continuous Random Variable Probabilities add up to 1 Individual probabilities are between 0 and 1

20 Definitions Mean or Expected Value Variance Standard Deviation These are of great practical importance

21 Expected Value Expected value is everywhere, we are using it regularly and very frequently in our lives Examples: Return on investments Insurance industry Product development/quality control Gambling

22 Binomial Distribution Conditions: Fixed number of trials Two outcomes Independent trials Probability is constant Example: Think of multiple choice test

23 Binomial Distribution Formula and Examples CURAIDS problem from the homework Credit Extension at a local furniture shop Airlines Booking S&P 500 rise year over year Babies born in a year to 10 couples

24 Binomial Distribution Mean of a Binomial Distribution Sigma of a Binomial Distribution Example: S&P 500 Airlines example Mean and standard deviation for all the Binomial Distribution examples given in class

25 Poisson Distribution Characteristics: Events happening randomly over a period of time or unit of space Average number of random events per unit is fixed Random events are independent of each other Random variable x represents the number times the random event happening per unit of time or space

26 Poisson Distribution Examples: Texting Customer arrivals at a local restaurant Emergency room arrival for texting while driving