Material Taken From:

Mathematicsfor 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 ProbabilityType Definition FormulaMutually 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 ProbabilityType Definition FormulaMutually 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)

)()()( BPAPBAP

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. Scotlandii. Walesiii. Scotland or Wales

Laws of ProbabilityType Definition FormulaMutually 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(AB) = P(A) + P(B) – P(A∩B)

Conditional Probability

Independent Events

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

)()()()( BAPBPAPBAP

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 ProbabilityTen children played two tennis matches each.

Child First Match Second Match

1 Won Won2 Lost Won3 Lost Won4 Won Lost5 Lost Lost6 Won Lost7 Won Won8 Won Won9 Lost Won10 Lost Won

What is the probability that a child won his first match, if it is known that he won his second match?

Laws of ProbabilityType Definition FormulaMutually 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(AB) = 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 pizzab) 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 fruitsb) Likes bananas given that he/she likes pineapplesc) 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 chickenb) was taken from the top shelf given that it is chicken

Independent Events

• If one student in the class was born on June 1st can another student also be born on June 1st?

• If you roll a die and get a 6, can you flip a coin and get tails?

Section 14L – Independent Events

Laws of ProbabilityType Definition FormulaMutually 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(AB) = 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 occurrence 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 exclusiveb) 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