Agenda

1. Agenda • Informationer • Uformel evaluering • Spørgeskema • Status på projekt 1 • Opsamling fra sidst • Sandsynlighedsregning • Sandsynlighedsfordelinger • Definitioner • Bedst på Nettet case! • Dagens øvelser Uformel evaluering • Godt: Tid til spm. og Gennemgang af formler • Mindre godt: Alt for meget matematik • Anderledes: Mere sp.skm.

2. The complement of an event • The complement of an event A consists of all outcomes in the sample space that are not in A. • The probabilities of A and of A’ add to 1 • P(A’) = 1 – P(A).

3. Probability of the Union of Two Events • Addition Rule: • For the union of two events, • P(A or B) = P(A) + P(B) – P(A and B) • If the events are disjoint, P(A and B) = 0, so • P(A or B) = P(A) + P(B) + 0

4. Probability of the Intersection of Two Events • Multiplication Rule: For the intersection of two independent events, A and B, P(A and B) = P(A) x P(B)

5. Calculation of probabilities • A=correct answer. • Probability of correct, P(A)=0,2. • Probability of 3 correct? 0,2 x 0,2 x 0,2 = 0,008. • Probability of less than 3 correct? 1- 0,008 = 0,992. • Probability of 3 at least 2 questions correctly? • P(0,008) + 3 x P(0,032)= 0,104

6. Learning Objectives • Probability distributions for discrete random variables • Mean of a probability distribution • Summarizing the spread of a probability distribution

7. Learning Objective 1:Probability Distribution • A random variable is a numerical measurement of the outcome of a random phenomenon. • The probability distributionof a random variable specifies its possible values and their probabilities.

8. Learning Objective 1:Random Variable • Use letters near the end of the alphabet, such as x, to symbolize • Variables • A particular value of the random variable • Use a capital letter, such as X, to refer to the random variable itself. Example: Flip a coin three times • X=number of heads in the 3 flips; defines the random variable • x=2; represents a possible value of the random variable

9. Learning Objective 2:Probability Distribution of a Discrete Random Variable • A discrete random variableX has separate values (such as 0, 1, 2, 3, 4, …) as its possible outcomes. • Its probability distribution assigns a probability P(x) to each possible value x: • For each x, the probability P(x) falls between 0 and 1 • The sum of the probabilities for all the possible x values equals 1

10. Learning Objective 2:Example • What is the probability of less than two home runs? • What is the probability of three home runs? • What is the probability of at least four home runs? • What is the probability of more than one, but less than four home runs? • What is the mean number of home runs?

11. Learning Objective 3:Example • Find the mean of this probability distribution. The mean: = 0(0.23) + 1(0.38) + 2(0.22) + 3(0.13) + 4(0.03) + 5(0.01) = 1.38

12. Learning Objective 3:The Mean of a Discrete Probability Distribution • The mean of a probability distribution for a discrete random variable is where the sum is taken over all possible values of x. • The mean of a probability distribution is denoted by the parameter, µ. • The mean is a weighted average; values of x that are more likely receive greater weight P(x)

13. Learning Objective 4:The Standard Deviation of a Probability Distribution The standard deviation of a probability distribution, denoted by the parameter, σ, measures its spread. • Larger values of σ correspond to greater spread. • Roughly, σ describes how far the random variable falls, on the average, from the mean of its distribution.

14. Learning Objectives • The Binomial Distribution • Conditions for a Binomial Distribution • Probabilities for a Binomial Distribution • Examples using Binomial Distribution • Do the Binomial Conditions Apply?

15. Learning Objective 1:The Binomial Distribution • Each observation is binary: It has one of two possible outcomes. • Examples: • The user clicks or does not click a link. • The user has an iPhone or he does not • The user buys a commodity in the web shop or he does not

16. Learning Objective 2:Conditions for the Binomial Distribution • Each of n trials has two possible outcomes: “success” or “failure”. • Each trial has the same probability of success, denoted by p. • The ntrials are independent. • The binomial random variable X is the number of successes in the n trials.

17. Learning Objective 3:Probabilities for a Binomial Distribution • Denote the probability of success on a trial by p. • For n independent trials, the probability of x successes equals:

18. Learning Objective 3:Probabilities for a Binomial Distribution • Excel har funktionen BINOMIAL.FORDELING • Man indtaster • Antal gunstige, f.eks. antal kroner • Det samlede antal forsøg, f.eks. antal kast • Sandsynligheden for et gunstigt udfald, f.eks. 0,5 • 0 (for at få en punktsandsynlighed)

19. Learning Objective 4:Binomial Example • Bogens eksempel 1 vedr. The Binomial Probability Distribution. • The American Heart Association claims that only 10% of adults over 30 can pass the minimum fitness requirement that is established by them. • Suppose that four adults, n=4, are randomly selected and given the fitness test. • Let X be the number of the four who pass the test. • What is the probability distribution for x?

20. Learning Objective 5:Do the Binomial Conditions Apply? • Before using the binomial distribution, check that its three conditions apply: • Binary data (success or failure). • The same probability of success for each trial (denoted by p). • Independent trials.