Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, 2004

Warm Up In a group of 108 people in an art gallery 60 liked the pictures, 53 liked the sculpture and 10 liked neither. What is the probability that a person chosen at random liked the pictures but not the sculpture?

BrainPop Video – Compound Events Section 14K – Laws of Probability Sometimes events can happen at the same time. Sometimes you will be finding the probability of event A or event B happening. Sometimes you will be finding the probability of event A and event B happening. Today:

Laws of Probability TypeDefinitionFormula Mutually Exclusive Events Combined Events (a.k.a. Addition Law) Conditional Probability Independent Events

Mutually Exclusive Events A bag of candy contains 12 red candies and 8 yellow candies. Can you select one candy that is both red and yellow?

Laws of Probability TypeDefinitionFormula Mutually Exclusive Events events that cannot happen at the same time P(A ∩ B) = 0 P(A  B) = P(A) + P(B) Combined Events (a.k.a. Addition Law) Conditional Probability Independent Events

P(either A or B) = P(A) + P(B) Mutually Exclusive Events

Example 1) Of the 31 people on a bus tour, 7 were born in Scotland and 5 were born in Wales. a)Are these events mutually exclusive? b)If a person is chosen at random, find the probability that he or she was born in: i.Scotland ii.Wales iii.Scotland or Wales

Laws of Probability TypeDefinitionFormula Mutually Exclusive Events events that cannot happen at the same time P(A ∩ B) = 0 P(A  B) = P(A) + P(B) Combined Events (a.k.a. Addition Law) events that can happen at the same time P(A  B) = P(A) + P(B) – P(A∩B) Conditional Probability Independent Events

P(either A or B) = P(A) + P(B) – P(A and B) Combined Events

Example 2) 100 people were surveyed: 72 people have had a beach holiday 16 have had a skiing holiday 12 have had both What is the probability that a person chosen has had a beach holiday or a ski holiday?

Example 3) If P(A) = 0.6 and P(A  B) = 0.7 and P(A  B) = 0.3, find P(B).

Conditional Probability Ten children played two tennis matches each. ChildFirst MatchSecond Match 1Won 2LostWon 3LostWon 4 Lost 5 6WonLost 7Won 8 9LostWon 10LostWon What is the probability that a child won his first match, if it is known that he won his second match?

Laws of Probability TypeDefinitionFormula Mutually Exclusive Events events that cannot happen at the same time P(A ∩ B) = 0 P(A  B) = P(A) + P(B) Combined Events (a.k.a. Addition Law) events that can happen at the same time P(A  B) = P(A) + P(B) – P(A∩B) Conditional Probability the probability of an event A occurring, given that event B occurred P (A | B) = P (A ∩ B) P (B) Independent Events

Example 4) In a class of 25 students, 14 like pizza and 16 like iced coffee. One student likes neither and 6 students like both. One student is randomly selected from the class. What is the probability that the student: a)likes pizza b)likes pizza given that he/she likes iced coffee?

Example 5) In a class of 40, 34 like bananas, 22 like pineapples and 2 dislike both fruits. If a student is randomly selected find the probability that the student: a)Likes both fruits b)Likes bananas given that he/she likes pineapples c)Dislikes pineapples given that he/she likes bananas

Example 6) The top shelf of a cupboard contains 3 cans of pumpkin soup and 2 cans of chicken soup. The bottom self contains 4 cans of pumpkin soup and 1 can of chicken soup. Lukas is twice as likely to take a can from the bottom shelf as he is from the top shelf. If he takes one can without looking at the label, determine the probability that it: a)is chicken b)was taken from the top shelf given that it is chicken

Independent Events If one student in the class was born on June 1 st can another student also be born on June 1 st ? If you roll a die and get a 6, can you flip a coin and get tails? Section 14L – Independent Events

Laws of Probability TypeDefinitionFormula Mutually Exclusive Events events that cannot happen at the same time P(A ∩ B) = 0 P(A  B) = P(A) + P(B) Combined Events (a.k.a. Addition Law) events that can happen at the same time P(A  B) = P(A) + P(B) – P(A∩B) Conditional Probability the probability of an event A occurring, given that event B occurred P (A | B) = P (A ∩ B) P (B) Independent Eventsoccurrence of one event does NOT affect the occurrence of the other P(A ∩ B) = P(A) P(B)

Find p if: a)A and B are mutually exclusive b)A and B are independent P (A) = ½ P (B) = 1/3 and P(A  B) = p Example 7)

Homework Worksheet: 16I.1 #1-4 all 16I.2 #1, 3, 5, 6, 7 16J