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?
Section 14K – Laws of Probability
• BrainPop Video
– Compound Events
Today:
• 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.
Laws of Probability
Type
Mutually Exclusive
Events
Combined Events
(a.k.a. Addition
Law)
Conditional
Probability
Independent Events
Definition
Formula
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
Type
Definition
Mutually Exclusive
Events
events that cannot
happen at the
same time
Combined Events
(a.k.a. Addition
Law)
Conditional
Probability
Independent Events
Formula
P(A ∩ B) = 0
P(A  B) = P(A) + P(B)
Mutually Exclusive Events
P( A  B)  P( A)  P( B)
P(either A or B) = P(A) + P(B)
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
Type
Definition
Mutually Exclusive
Events
events that cannot
happen at the
same time
Combined Events
(a.k.a. Addition
Law)
events that can
happen at the
same time
Conditional
Probability
Independent Events
Formula
P(A ∩ B) = 0
P(A  B) = P(A) + P(B)
P(AB) = P(A) + P(B) – P(A∩B)
Combined Events
P( A  B)  P( A)  P( B)  P( A  B)
P(either A or B) = P(A) + P(B) – P(A and B)
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.
Child
1
2
3
4
5
6
7
8
9
10
First Match Second Match
Won
Lost
Lost
Won
Lost
Won
Won
Won
Lost
Lost
Won
Won
Won
Lost
Lost
Lost
Won
Won
Won
Won
What is the probability
that a child won his
first match, if it is
known that he won his
second match?
Laws of Probability
Type
Mutually
Exclusive Events
Definition
Formula
events that cannot
P(A ∩ B) = 0
happen at the same
P(A  B) = P(A) + P(B)
time
Combined Events
(a.k.a. Addition
Law)
events that can
P(AB) = P(A) + P(B) – P(A∩B)
happen at the same
time
Conditional
Probability
the probability of
an event A
occurring, given
that event B
occurred
Independent
Events
P (A | B) = P (A ∩ B)
P (B)
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
Section 14L – Independent Events
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?
Laws of Probability
Type
Definition
Mutually Exclusive events that cannot
Events
happen at the same
time
Formula
P(A ∩ B) = 0
P(A  B) = P(A) + P(B)
P(AB) = P(A) + P(B) – P(A∩B)
Combined Events
(a.k.a. Addition
Law)
events that can
happen at the same
time
Conditional
Probability
the probability of an P (A | B) = P (A ∩ B)
event A occurring,
P (B)
given that event B
occurred
Independent Events occurrence of one
event does NOT
affect the
occurrence of the
other
P(A ∩ B) = P(A) P(B)
Example 7)
P (A) = ½
P (B) = 1/3
and P(A  B) = p
Find p if:
a) A and B are mutually exclusive
b) A and B are independent
Homework
• Worksheet:
• 16I.1 #1-4 all
• 16I.2 #1, 3, 5, 6, 7
• 16J