Probability of Compound Events - There are (2) types of compound events: Independent Events – involves two or more events in which the outcome of one event DOES NOT affect the outcome of any other events Examples: roll dice, coin flip, problems with replacement P(A and B) = P(A) x P(B) Dependent Events- involves two or more events in which the outcome of one event DOES affect the outcome of any other events Examples: deck of cards, selecting item from container, problems without replacement P(A and B) = P(A) x P(B following A)

Probability of Compound Events Example 1: Roll a dice. What is the probability of rolling back to back sixes? P(6,then 6) = (1/6)(1/6) = (1/36)

Probability of Compound Events Example 2: Roll a dice. What is the probability of rolling back to back evens? P(even,then even) = (3/6)(3/6) = (1/2)(1/2) = (1/4)

Drawing Without Replacement Sampling without replacement means that once one object is drawn it doesn’t go back into the pool. We often sample without replacement, which doesn’t matter too much when we are dealing with a large population. However, when drawing from a small population, we need to take note and adjust probabilities accordingly. Drawing without replacement is just another instance of working with conditional probabilities.

Probability W/O Replacement Example 3: Best Buy is having an IPOD giveaway. They put all the IPOD Shuffles in a bag. Customers may choose an IPOD without looking at the color. Inside the bag are 4 orange, 5 blue, 6 green, and 5 pink IPODS. If Maria chooses one IPOD at random and then her sister chooses one IPOD at random, what is the probability they will both choose an orange IPOD?

Probability W/O Replacement Example 3: Best Buy is having an IPOD giveaway. They put all the IPOD Shuffles in a bag. Customers may choose an IPOD without looking at the color. Inside the bag are 4 orange, 5 blue, 6 green, and 5 pink IPODS. If Maria chooses one IPOD at random and then her sister chooses one IPOD at random, what is the probability they will both choose an orange IPOD? P(orange,orange) = 4/20 x 3/19 = 3/95 or 3.2%

Probability W/O Replacement Example 4: Deck of Cards. What is the probability of drawing 2 hearts (without replacement)? P(Heart and Heart) = (13/52)(12/51) = (1/17)

Probability Examples Practice: (1) Wyatt has four $1 bills in his wallet and three $10 bills in his wallet. What is the probability he will reach into his wallet twice and pull out $20 in total? (2) A bag contains 3 green and 2 purple marbles. What is the probability of drawing two purple marbles in a row from the bag if the first marble is not replaced?

The First Three Rules of Working with Probability (MAKE A PICTURE) The most common kind of picture to make is called a Venn diagram. We will see Venn diagrams in practice shortly…

Example #1 A survey of 64 informed voters revealed the following information: 45 believe that Elvis is still alive 49 believe that they have been abducted by space aliens 42 believe both of these things

General Addition Rule: P(A or B) = P(A) + P(B) – P(A and B) For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B) (On the Formula Sheet) The following Venn diagram shows a situation in which we would use the general addition rule:

Example #2 A survey of 88 faculty and graduate students at the University of Florida's film school revealed the following information: 51 admire Moe 49 admire Larry 60 admire Curly 34 admire Moe and Larry 32 admire Larry and Curly 36 admire Moe and Curly 24 admire all three of the Stooges

51 admire Moe 49 admire Larry 60 admire Curly 34 admire Moe and Larry 32 admire Larry and Curly 36 admire Moe and Curly 24 admire all three of the Stooges

Bernoulli Trials The basis for the probability models we will examine in this chapter is the Bernoulli trial. We have Bernoulli trials if: there are two possible outcomes (success and failure). the probability of success, p, is constant. the trials are independent.

The Binomial Model A Binomial model tells us the probability for a random variable that counts the number of successes in a fixed number of Bernoulli trials. Two parameters define the Binomial model: n, the number of trials; and, p, the probability of success. We denote this Binom(n, p).

The Binomial Model (cont.) In n trials, there are ways to have k successes. Read nCk as “n choose k,” and is called a combination. Note: n! = n x (n – 1) x … x 2 x 1, and n! is read as “n factorial.”

The Binomial Model (cont.) Binomial probability model for Bernoulli trials: Binom(n,p) n = number of trials p = probability of success q = 1 – p = probability of failure X = number of successes in n trials