Download Theoretical Probability

THEORETICAL PROBABILITY
Lesson 16
WARM UP

Name the property illustrated.

3+0=3

2 + -2 = 0

2(x + 5) = 2x + 10

2 + (3 + 5) = (2 + 3) + 5
WARM UP- SOLUTIONS

Name the property illustrated.

3+0=3


2 + -2 = 0


Inverse
2(x + 5) = 2x + 10


Identity
Distributive Property
2 + (3 + 5) = (2 + 3) + 5

Associative Property
WHAT IS PROBABILITY?

The probability of an outcome is a ratio of
number of ways the desired outcome can occur
the total number of possible outcomes.

It is the likelihood of an event happening.
Examples
The probability of rolling a 2 on a die is 1/6
 The probability of a heads is ½
 The probability of drawing a king from a standard deck is
4/52 = 1/13

EXAMPLE 1

Kylee wrote the names of the months of the year
on slips of paper and put them in a box. If she
picks out one slip, find the following probabilities.

P(picking a month that has exactly 4 letters)

P(picking a month that begins with a vowel)

P(picking a month that has at least 3 letters)
EXAMPLE 1- SOLUTIONS
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January
February
March
April
May
June
July
August
September
October
November
December
P(picking a month that has exactly 4
letters)
There are 2 months that have 4 letters:
June, July

2/12 = 1/6
P(picking a month that begins with a
vowel)
There are that 3 months that start with a
vowel: April, August, October
3/12 = 1/4
 P(picking a month that has at least 3
letters)
All of the months have at least 3 letters
12/12 = 1

EXAMPLE 2

Patrick has a bag of m&ms. It contains 8 blue,
15 green, 3 orange and 12 red. Patrick will pick a
candy out of the bag without looking. Find each
probability.

P(green)

P(brown)

P(orange or green)
EXAMPLE 2- SOLUTIONS

Patrick has a bag of m&ms. It contains 8 blue,
15 green, 3 orange and 12 red. Patrick will pick a
candy out of the bag without looking. Find each
probability.
P(green)
There are 38 total candies, 15 are green.
P(green) = 15/38
 P(brown)
There are no brown candies.
P(brown) = 0/38 = 0
 P(orange or green)
3 + 15 = 18
18/38 = 9/19

EXAMPLE 3

Patrick has a bag of m&ms. It contains 8 blue,
15 green, 3 orange and 12 red. Patrick will pick a
candy out of the bag without looking.

Which color is he most likely to pick from the bag?

Which color is he least likely to pick from the bag?
EXAMPLE 3- SOLUTIONS

Patrick has a bag of m&ms. It contains 8 blue,
15 green, 3 orange and 12 red. Patrick will pick a
candy out of the bag without looking.
Which color is he most likely to pick from the bag?
Since there are more green than any other color, green
is the most likely color to occur.

Which color is he least likely to pick from the bag?
Since there are only 3 orange, and more of every other
color, orange is least likely to occur.

INDEPENDENT VS DEPENDENT EVENTS

Independent events do not effect each other.
Events “with replacement”, such as drawing a card
and placing it back in the deck.
 Rolling dice (a 3 on the first roll has no effect on the
second roll).


Dependent events effect each other.
Events “without replacement”, such as dealing cards.
 Choosing students to be on a team.

COMBINED PROBABILITIES

When you want to find the probability of more
than one event at a time, multiply each
individual probability together.

Examples:
Find the probability of rolling a 3 followed by a 2.
Each of these events are independent. The probability of
rolling a 3 is 1/6, the probability of tolling a 2 is 1/6 so the
probability of one followed by the next is:
(1/6)(1/6)= 1/36

Find the probability of drawing a king followed by an ace.
These events are dependent. The probability of drawing a
king is 4/52. Once the king is drawn there are only 51 cards
left, so the probability of an ace is 4/51. The probability of
both is
(4/52)(4/51) = 16/2652 = 4/663

EXAMPLE 4
Determine if the following events are
independent or dependent.
 Then find the probability


There are 6 questions on a multiple choice test. Each
question has 4 answer choices. What is the
probability of a person guessing all of them correctly?

There are 6 questions on a matching test with exactly
one answer paired with each question. What is the
probability that a person who guesses on all five
questions will answer them all correctly?
EXAMPLE 4- SOLUTIONS
Determine if the following events are
independent or dependent.
 Then find the probability


There are 6 questions on a multiple choice test. Each
question has 4 answer choices. What is the
probability of a person guessing all of them correctly?
Each guess is independent. Getting one right does
not effect the chances of getting the next one
right.
1 1 1 1 1 1  1 6
1
         
4 4 4 4 4 4  4  4096
EXAMPLE 4- SOLUTIONS
Determine if the following events are
independent or dependent.
 Then find the probability


There are 6 questions on a matching test with exactly
one answer paired with each question. What is the
probability that a person who guesses on all five
questions will answer them all correctly?
These events are not independent. Once you get
the first question right there are only 5 more to
choose from.
1 1 1 1 1 1 1
      
6 5 4 32 1 720
EXAMPLE 5


There are 25 students in a class. 12 of them are
girls and 13 are boys. If all of their names are
placed in a hat, what is the probability that a girl
will be drawn first followed by a boy?
What is the probability that 2 boys names will be
drawn?
EXAMPLE 5- SOLUTION
There are 25 students in a class. 12 of them are
girls and 13 are boys. If all of their names are
placed in a hat, what is the probability that a girl
will be drawn first followed by a boy?
These are dependent events.
12 13  156  13 
      
25 24  600  50 

What is the probability that 2 boys names will be
drawn?
These
are dependent events.


12 11  132  11 
      
25 24  600  50 
EXAMPLE 6

The spinner is spun once.

What is the probability that the
number spun is divisible by 3?

What is the probability that the
number spun is less that 5?

What is the probability that the
number spun is even or a 5?
EXAMPLE 6- SOLUTIONS

The spinner is spun once.

What is the probability that the
number spun is divisible by 3?
3 and 6 are divisible by 3
2/8 = 1/4

What is the probability that the
number spun is less that 5?
1,2,3,4 are less than 5
There are 8 values on the
spinner, each one is the 4/8 = 1/2
same size so they each
 What is the probability that the
have the same probability of
number spun is even or a 5?
occurring: 1/8
2,4,6,8,5
5/8
FUNDAMENTAL COUNTING PRINCIPLE
A pizza shop wants to see how many outcomes
can be made with the following options.
 Crust: pan, thin. Cheese: mozzarella, Parmesan;
Toppings: Pepper, Ham, Sausage
 How many different pizzas can be made choosing
1 crust, 1 cheese and 1 topping.

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2 crust x 2 cheese x 3 toppings = 12 possible outcomes
Tree diagrams can be used to show the outcome
set of a situation.
EXAMPLE 7
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The first 3 questions on a history test are
true/false. Make a tree diagram to show how
many different ways the 3 questions can be
answered. (use T for true and F for false )
EXAMPLE 7- SOLUTION

The first 3
questions on a
True
history test are
true/false. Make a
tree diagram to
show how many
different ways the
3 questions can be False
answered.
True
False
True
False
True
False
True
False
True
False
True
False
**Notice there are 2 x 2 x 2 = 8 ways that the 3 questions can be
answered
EXAMPLE 8

A coin is tossed up in the air 4 times. Makes a
tree diagram to show how many different ways
all 4 tosses can land. (Use H for heads and T for
tails.)
Heads
Heads
EXAMPLE 8- SOLUTION

A coin is tossed up
in the air 4 times.
Makes a tree
diagram to show
how many
different ways all
4 tosses can land.
Tails
Heads
Heads
Tails
Tails
Heads
Heads
Tails
Tails
Heads
Tails
Tails
Heads
Tails
Heads
Tails
Heads
Tails
Heads
Tails
Heads
Tails
Heads
Tails
Heads
Tails
Heads
Tails
EXAMPLE 9

Using the digits 0-9, how many different 4 digit
numbers are possible? (Repetition of digits is
allowed.)
EXAMPLE 9- SOLUTION

Using the digits 0-9, how many different 4 digit
numbers are possible? (Repetition of digits is
allowed.)
(10)(10)(10)(10) = 10000
EXAMPLE 10

Using the digits 0-9, how many different 4 digit
numbers are possible if the first digit cannot be
zero? (Repetition of digits is allowed.)
EXAMPLE 10- SOLUTION

Using the digits 0-9, how many different 4 digit
numbers are possible if the first digit cannot be
zero? (Repetition of digits is allowed.)
(9)(10)(10)(10) = 9000
EXAMPLE 11

The Math Club has 12 girls, 8 boys, and 4 adults
chaperones going on a field trip. How many
different groups of 1 girl, 1 boy, and 1 adult are
there?
EXAMPLE 11- SOLUTION

The Math Club has 12 girls, 8 boys, and 4 adults
chaperones going on a field trip. How many
different groups of 1 girl, 1 boy, and 1 adult are
there?
(12)(8)(4) = 384
EXAMPLE 12

Arizona license plates consist of 3 digits followed
by 3 letters. How many different license are
possible? (Repetition of digits and letters are
allowed.) Set up, but do not multiply out.
EXAMPLE 12- SOLUTION

Arizona license plates consist of 3 digits followed
by 3 letters. How many different license are
possible? (Repetition of digits and letters are
allowed.) Set up, but do not multiply out.
(10)(10)(10)(26)(26)(26)
EXAMPLE 13

A particular shirt comes in 2 colors, 2 styles, and
4 sizes. The following table shows all the choices.
Color
Style
Size
Black
Crew neck
Small
Gold
V-neck
Medium
Large
X-Large

How many different shirts are possible?
EXAMPLE 13- SOLUTION

A particular shirt comes in 2 colors, 2 styles, and
4 sizes. The following table shows all the choices.
Color
Style
Size
Black
Crew neck
Small
Gold
V-neck
Medium
Large
X-Large
How many different shirts are possible?
(2)(2)(4) = 16 different shirts

EXAMPLE 14

A meal at a certain restaurant includes one type
of meat, potato, and vegetable. The following
table show all the choices.
Main
Side
Vegetable
Beef
Baked Potato
Peas
Fish
Scalloped
Carrots
French Fries

How many meals are possible?
EXAMPLE 14- SOLUTION

A meal at a certain restaurant includes one type
of meat, potato, and vegetable. The following
table show all the choices.
Main
Side
Vegetable
Beef
Baked Potato
Peas
Fish
Scalloped
Carrots
French Fries
How many meals are possible?
(2)(3)(2) = 12 different meals
