Conditional Probabilities Multiplication Rule Independence Conditional Probability Conditional

Conditional Probabilities Multiplication Rule Independence

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) A = odd B = even C = bigger than 4 D=5 Find: P(D) P(D|A) P(D|C) P(B|A)

Conditional Probability • Dog example from last time • Find each – P(SH|Med) – P(Med|SH) – P(Large|Not. SH) – P(Not. SH|small) Short Hair NOT Short Hair Total Small 3 4 7 Medium 5 3 8 Large 5 3 8 Total 13 10 23

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

Conditional Probability • Formula: P(B|A) = Revisit a couple of card or dice examples. 227 ex 5. 17

The Multiplication Rule • Conditional Probability is P(B|A) = Solve for P(A&B) Pg 230 Ex 5. 20 Pg 230 Ex 5. 21

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

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

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)

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)…. )

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?

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.

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

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

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

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

Permutations of Non-Distinct Items Where n = n 1 + n 2 + …. + nk Pg 249 ex 5. 41

Examples • Pg 250 ex 5. 42 • Pg 251 ex 5. 43 • Lottery activity