1. Conditional Probabilities Multiplication Rule Independence

2. Conditional Probability Conditional probability: P(B|A) “the probability of B given A” (what is the probability of B happening when you know that A has already happened) Dice (6 sided) Find:P(D) A = odd B = evenP(D|A) C = bigger than 4P(D|C) D = 5P(B|A)

3. Conditional Probability Dog example from last time Find each – P(SH|Med) – P(Med|SH) – P(Large|NotSH) – P(NotSH|small) Short HairNOT Short HairTotal Small347 Medium538 Large538 Total131023

4. Conditional Probability Cards A = red B = heart C = face card D = even (2, 4, 6, 8, 10) E = king Find the probability of each P(B|A) P(E|C) P(not E|C) P(C|E) P(C| not E)

5. Conditional Probability Formula: P(B|A) = Revisit a couple of card or dice examples. 227 ex 5.17

6. The Multiplication Rule Conditional Probability is P(B|A) = Solve for P(A&B) Pg 230 Ex 5.20 Pg 230 Ex 5.21

7. Independence Statistical independence: the probability of one event (B) does not depend on the other event’s (A) occurrence. A and B are independent if P(B|A) = P(B) Pg 228 ex 5.18 Pg 229 ex 5.19 Magazine Excerpt

8. Special Multiplication Rule IF events A, B, C…. are independent events, then P(A & B & C &…) = P(A)*P(B)*P(C)*… You play a game using a standard dice where if you roll a 6 first, then a 2, and then a 4 you will $800. What is the probability you win the game? If you roll a dice 3 times, which has a better chance of occurring in the given order: 1, 1, 1 or 2, 3, 6

9. Mutually Exclusive Vs. Independent Events Mutually exclusive and independent are often confused terms…here is a mathematical reasoning why they are NOT the same thing: Mutually Exclsive: P(A & B) = 0 Independent Events: P(A & B) = P(A)P(B)

10. Mutually Exclusive Vs. Independent Events Here is an example of each: You have a die that has 6 sides and each side is a different color. The 1 is red, the 2 is orange…you also have cards numbered 1-10. Mutually exclusive: rolling the die once and rolling a 1 AND the orange side (not possible so P = 0). Independence: rolling a 1 and drawing a 7 from the cards (the cards don’t care what you rolled on the die and the die doesn’t care what card you drew…P=(1/6)*(1/10)=(1/60)….)

11. Counting Rules Use counting rules to determining the number of ways something can happen. Sandwich scenario and tree diagram The Basic Counting Rule: Telephone #’s: (first # can’t be 0) JFK…FDR…How many three letter initials are possible?

12. Factorial, Permutation, and Combination K! (k factorial) 6! = 6*5*4*3*2*1 = 720 Permutation: a permutation of r objects from a collection of n objects is any ORDERED arrangement of r of the n objects. Combination: a combination of r objects from a collection of n objects is any UNORDERED arrangement of r of the n objects.

13. Permutations and Combinations Examples: ORDER matters! – President, vice-president, secretary – 1 st place, 2 nd place, 3 rd place – Combination of a lock – Computer Programming – Batting order of a baseball team – Telephone numbers – License plate numbers Examples: Order does NOT matter – Picking an advisory board (no “positions”) – Picking a sample of 5 students from the class (does it matter if you pick Jimmy first or third?) – A grape, apple and banana fruit salad – Lottery numbers – BINGO – You are picking 3 dips of ice cream

14. Permutations The number of permutations of r objects from a collection of n objects is given by: nPr =

15. Combinations The number of combinations of r objects from a collection of n objects is given by: nCr =

16. Calculators See if you have a “STAT” or “PROB” or “MATH” or “PRB” or “PROBABILITY” button…. This will take you to the combination, permutation, and possibly factorial keys. Pg 246 ex 5.36 Pg 247 ex 5.37

17. Permutations of Non-Distinct Items Where n = n 1 + n 2 + ….+ n k Pg 249 ex 5.41

18. Examples Pg 250 ex 5.42 Pg 251 ex 5.43 Lottery activity