Def: The intersection of A and B is the set of all elements common to both set A and set B

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1 Def: Sample Space the set of all possible outcomes Def: Element an item in the set Ex: The number "3" is an element of the "rolling a die" sample space Main concept write in Interactive Notebook Intersection: "AND" Union: "OR" Complement: What do these look like? Def: The intersection of A and B is the set of all elements common to both set A and set B Def: The union of A and B is the set of all elements in A PLUS all elements in set B. "NOT" Def: The complement of set A is the set of all elements NOT in A. (write and draw these on the foldable) 1

2 : The chances of something happening. How do you calculate it? Basic : the number of ways of what you WANT = WANT the TOTAL number of possible ways TOTAL Theoretical * What SHOULD happen based on equally likely outcomes Experimental * What ACTUALLY happens based on experiments or past events write these on the foldable Experimental : Partner Up: 1) Each group take a die 2) Pick a number between 1 and 6 3) Roll the die 10 times and count how many times the number you "WANT" shows up 4) Calculate the experimental probability of getting that number. write your example on the foldable 2

3 Theoretical Example: What is the probability of rolling a "3" on a standard six sided die? Answer: 1 value that I WANT = 1 =.1667 = 16.67% 6 TOTAL possible values 6 write your example on the foldable Question 1: What is the probability of rolling a 3 or a 4 on a standard six sided die? Answer: 2/6 = 1/3 =.333 = 33% Question 2: What is the probability of rolling an even number on a standard six sided die? Answer: 3/6 = 1/2 =.5 = 50% Question 3: What is the probability getting tails on a standard two sided coin? Answer: 1/2 =.5 = 50% Homework: Experimental and Theoretical 3

4 Standard deck of cards: 52 cards total (not including jokers) 4 suits (black spade, black club, red heart, red diamond) Each suit has 13 cards (A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K) Face cards are those with "faces" (J, Q, K) total of 12 in the deck Example: What is the probability of picking a red card from a standard deck of cards? Answer: 26 Red cards that I WANT / 52 TOTAL cards = 26/52 = 1/2 =.5 = 50% Question 4: What is the probability of picking an Ace from a standard deck of cards? Answer: 4/52 = 1/13 =.0769 Question 5: What is the probability of picking a spade from a standard deck of cards? Answer: 13/52 = 1/4 =.25 Question 6: What is the probability of picking a diamond from a standard deck of cards? Answer: 13/52 = 1/4 =.25 Question 7: What is the probability of picking a face card from a standard deck of cards? Answer: 12/52 =

5 Distribution Probabilities are assigned a decimal value between 0 and 1. 0 = impossible 1 = guaranteed to happen Main Concept write in INB Mutually Exclusive Two events that cannot occur at the same time. Example: Being a red card and a black card if two events are mutually exclusive, then P(A and B) = 0 Main Concept write in INB write on foldable Addition Principle of : if looking for the probability of either A or B occurring, ADD A + B General Rule: P(A or B) = P(A) + P(B) P(A and B) color venn diagram on foldable Example: What is the probability of getting a 3 or a 4 on a standard six sided die? Answer: P(3) = 1/6, P(4) = 1/6; P(3 OR 4) = 1/6 + 1/6 = 2/6 = 1/3 =.333 = 33.3% 5

6 Question 8: What is the probability of rolling a 1 or 2 on a standard six sided die? Answer: 1/3 =.333 Question 9: What is the probability of rolling a 1, 2, or a 3 on a standard six sided die? Answer: 3/6 = 1/2 =.5 NOTE: The sum of the probabilities of ALL possible outcomes is ALWAYS "1". Question 10: What is the probability of rolling an even OR an odd number on a standard six sided die? Answer: 1 write in INB Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring. > Ex: Landing on heads after tossing a coin AND rolling a 5 on a single 6 sided die. Go to: Math Goodies online 6

7 write on foldable Multiplication Principle of : If looking for the probability of both A and B occurring, MULTIPLY AB General Rule: P(A and B) = P(A) P(B) Example: What is the probability of getting a 3 AND then a 4 when rolling a single six sided die twice? Answer: P(3) = 1/6, P(4) = 1/6; P(3 AND 4) = 1/6 x 1/6 = 1/36 =.0278 = 2.78% Be Aware: In probabilities, order is always considered Example: What is the probability of getting a 3 AND a 4 when rolling two six sided dice? Answer: Two things could have happened: 1) Could have rolled a 3 and then a 4... P(3) = 1/6, P(4) = 1/6; P(3 AND 4) = 1/6 x 1/6 = 1/36 OR 2) Could have rolled a 4 and then a 3. P(4) = 1/6, P(3) = 1/6; P(4 AND 3) = 1/6 x 1/6 = 1/36 *Since it said: P(3 and 4) OR P(4 and 3) = ADD 1/36 + 1/36 = 2/36 = 1/18 =.0556 = 5.56% 7

8 Question 11: What is the probability of rolling a 5 AND a 6 when rolling two dice? Answer: 1/36 + 1/36 = 2/36 = 1/18 =.0556 Question 12: What is the probability of rolling all 6's when rolling three dice? Answer: 1/6 x 1/6 x 1/6 = 1/216 =.0046 Now...let's think about probabilities with and without replacement for a finite number of items. Take a deck of cards...there are 52 cards...that is a finite number (or limited number) of cards available. Each time you pick a card from the deck if you return the card to the deck before picking again (with replacement) it doesn't change your probability. BUT, if you pick a card and KEEP the card (without replacement) it CHANGES your chances of picking your next card. 8

9 Example: What is the probability of picking two Red cards from a standard deck: (a) with replacement? (b) without replacement? Still 26 Still 52 Red Cards Total Cards Red Cards Total Cards Answer: (a) P(First Red) = 26/52, P(Second Red after replacement) = 26/52 P(Red) AND P(Red) = 26/52 x 26/52 = 676/2704 = 1/4 =.25 = 25% Red Cards Total Cards Answer: (b) P(First Red) = 26/52, P(Second Red w/o replacement) = 25/51 Only 25 Red Only 51 Total Cards Left Cards Left in the pile P(Red) AND P(Red) = 26/52 x 25/51 = 650/2652 =.2451 = 24.51% Question 13: What is the probability of picking two Aces from a standard deck: (a) with replacement? Answer: 4/52 x 4/52 = 16/2704 = (b) without replacement? Answer: 4/52 x 3/51 = 12/2652 = Question 14: What is the probability of picking FOUR Face Cards from a standard deck: (a) with replacement? Answer: 12/52 x 12/52 x 12/52 x 12/52 = 20736/ = (b) without replacement? Answer: 12/52 x 11/51 x 10/50 x 9/49 = 11880/ =

10 Question 15: What is the probability of picking a Red and a Black card from a standard deck: (remember you could pick a Red and then a Black OR a Black and then a Red) (a) with replacement? Answer: R(26/52) x B(26/52) OR B(26/52) x R(26/52) = 676/ /2704 = 1352/2704 =.50 = 50% (b) without replacement? Answer: R(26/52) x B(26/51) OR B(26/52) x R(26/51) = 676/ /2652 = 1352/2652 =.5098 Quiz Time ( Quiz #1) 10

11 Two events are dependent if the outcome or occurrence of the first affects the outcome or occurrence of the second so that the probability is changed. Ex: A card is chosen at random from a standard deck of 52 playing cards. Without replacing it, a second card is chosen. What is the probability that the first card chosen is a queen and the second card chosen is a jack? Conditional the probability of a event given that (by assumption, presumption, assertion or evidence) another event has occurred Ex: The Titanic: First Second Third Crew Total Alive Dead Total P(A l B) "given that" 11

12 1. What percent of people died? First Second Third Crew Total Alive Dead Total What percent of people were crew members? 3. What percent of crew members died? 4. What percent of those who died, were crew mwmbers? 5. What percent of first class survived? Go to: Math Goodies online some additional things to think about. All the Now...what about if everything is NOT ALL what you wanted? Example: What is the probability of getting 4 Aces in a 5 card Poker hand? Answer: You have to think about ORDER ( always thinks about order) different ways it could happen A, A, A, A, Not A = 4/52 x 3/51 x 2/50 x 1/49 x 48/48 = A, A, A, Not A, A = 4/52 x 3/51 x 2/50 x 48/49 x 1/48 = A, A, Not A, A, A = 4/52 x 3/51 x 48/50 x 2/49 x 1/48 = A, Not A, A, A, A = 4/52 x 48/51 x 3/50 x 2/49 x 1/48 = Not A, A, A, A, A = 48/52 x 4/51 x 3/50 x 2/49 x 1/48 = the number of different ways that these combinations can happen is the same as ncr = 5 C 4 = 5 ways Notice that these are all the same, so order doesn't change the individual probability 12

13 a little more practice Question 17: A pair of dice is rolled. What is the probability of: (a) a pair of 5's (b) At least one 5 (c) A sum of 3 (d) A sum greater than 8 (e) A sum less than 7 Answer: 1/6 x 1/6 = 1/36 =.0278 Answer: 1/6 x 5/6 + 5/6 x 1/6 = 5/36 + 5/36 = 10/36 =.278 Answer: Roll: 2,1 or 1,2 = 1/36 +1/36 = 2/36 = 1/18 =.0556 Answer: Roll: 5,4 or 4,5 or 6,4 or 4,6 or 6,5 or 5,6 or 6,6 = 7 ways x 1/36 each = 7/36 =.1944 Answer: Roll: 1,1 or 1,2 or 1,3 or 1,4 or 1,5 or 2,1 or 2,2 or 2,3 or 2,4 or 3,1 or 3,2 or 3,3 or 4,1 or 4,2 or 5,1 = 15 ways x 1/36 each = 15/36 =.4167 a little more practice Question 18: A candy maker makes candies that come in the given proportions below. A single candy is picked at random. What is the probability that... color probability (a) the candy is blue? Answer: 0.2 blue.2 (b) the candy is white or blue? Answer: 0.4 (c) the candy is neither white nor pink? Answer: 0.6 red.1 (d) both candies are green? Answer: 0.2 x 0.2 = 0.04 white.2 (e) one is red and the other is pink? Answer: Red (0.1) x Pink (0.2) OR green.2 (f) neither candy is pink? Pink (0.2) x Red (0.1) = = 0.04 pink.2 Answer: 0.8 x 0.8 = 0.64 tan.1 13

14 Quiz Time Again ( Quiz #2) 14