More on Probability
by David Palay
Review
Evaluate:
3! ∗ 6!
3!
9! − 6!
9−6 !
5 𝑃4
8 𝐶7
Review
In how many ways can the debate team choose a
president and a secretary if there are 10 people on the
team?
Mrs. Marshall has 11 boys and 14 girls in her class.
A. How many ways can she select 2 girls to pass out a
snack?
B. How many ways can she select 5 boys to pass out new
books?
C. How many ways can she select 3 students to carry a
desk to the office?
Review
Definition of Experimental Probability:
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑃 𝐸 =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
Definition of Theoretical Probability:
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝐸 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟
𝑃 𝐸 =
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
More Review
The Law of Large Numbers says, roughly, that as
the number of trials increases,
Experimental Probability approaches Theoretical
Probability
Review
What is the probability that a point chosen
randomly from within the rectangle will be
within the shaded square?
2 in.
7 in.
4 in.
review
Rolls of 2 six sided dice
30
25
27
24
Total occurrences
25
20
15
10
5
16
14
10
8
7
3
4
9
6
4
0
2
5
6
7
8
9
10
11
12
Sum
The above chart represents the outcome of
150 rolls of 2d6. What is the experimental
probability of:
Rolling a 7?
Rolling a 10 or higher?
Rolling a 9 or Lower?
review
Hmmm… Something interesting there about 10
and higher vs. 9 and lower…
What do we get when we add them together?
Review Extra
Interesting…
What conclusion can we draw from this?
Compliments Rule
𝑃 𝐴 + 𝑃 𝐴𝑐 = 1
Combining events
𝑃 𝐴 ∪𝐵 =?
Venn Diagrams
Event A
New Symbols!
Name
Union
Loose
Terms
Or
Intersection
And
Compliment
But Not
Symbol
Example
∪
∩
𝐴 ∪𝐵
𝐴 ∩𝐵
𝑐
𝐴
𝑐
𝐴∩𝐵
𝐴∪𝐵
𝑐
𝐴
So how about these?
𝐴∪𝐵∪𝐶
𝐴𝑐 ∩ 𝐵
𝐴𝑐 ∪ 𝐵
𝐴∩𝐵 ∪ 𝐴∩𝐶
𝐴∩𝐵∩𝐶
𝑐
∩( 𝐴∩𝐵 ∪ 𝐴∩𝐶 )
Addition rule for unions
𝑃 𝐴∪𝐵 =𝑃 𝐴 +𝑃 𝐵 −𝑃 𝐴∩𝐵
The probability of A OR B is
the Probability of A
PLUS
the Probability of B
MINUS
the Probability of A AND B
Mutual Exclusion
Two events are mutually exclusive if:
• 𝑃 𝐴∩𝐵 =0
• There is no overlap
• There is no possibility of them happening
together
• No outcomes in
common
Change of Pattern
• In the following Venn Diagrams,
we often assume that two events,
A and B, constitute the entire
Universe. That is, there is
nothing outside of A and B.
Complimenting complex sets
What is the compliment of 𝐴 ∩ 𝐵?
𝐴𝑐 ∪ 𝐵𝑐
What about 𝐴 ∩ 𝐵 ∪ 𝐶 ’s compliment?
𝐴𝑐 ∪ (𝐵𝑐 ∩ 𝐶 𝑐 )
So, algorithm
Write these on the Board…
The Monty Hall Problem
• Monty Hall, the host of Let’s Make a Deal, used to
offer contestants this situation:
– There are three doors, behind one is a great prize
(Let’s go with new car for the sake of our example)
and behind the other two doors are fake prizes (goats,
is the usual example here)
– The contestant is told to pick one door, but not to open
it. The Host, Mr. Hall, then would open one of the two
other doors to reveal one of the goats.
– The contestant is now given the opportunity to switch
his/her choice to the as-yet untouched & unpicked
third door.
Extra Practice
The chart describes
the transportation
methods for 162
travellers.
If a traveller is selected at random, what is the probability
that:
P(ferry or train)
P(ferry or rental car)
P(train and ferry, or train and rental car)