Download Practice - SP Moodle

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
Topic 3 – Logic, Sets and
Probability
3.6 – Probability A
Basic Probability Theory
Probability
• The study of the chance of an event(s) happening
• The branch of mathematics that analyzes random
experiments
Random Experiment
• One in which we cannot predict the precise outcome
• Examples include; flipping a coin, rolling a die or
predicting who will will the next World Cup
• Although it is impossible to predict the outcome
precisely, it is possible to:
I. List the set of possible outcomes of the experiment
II. Decide how likely a particular outcome may be
Basic Probability Theory
Random Experiment
List the set of possible outcomes of the experiment:
• Coin Flip – {H, T} – P(H) or P(T)
• Die Roll – {1,2,3,4,5,6} – P(1), P(2), P(3) …
• Draw a Card – {1❤️, 2❤️, 3❤️…} – P(1❤️), P(2❤️) …
In the nomenclature P(E) the P stands for probability
while the E is the outcome or event. Thus P(E) is the
probability of a particular event occurring.
Basic Probability Theory
success
P(E) 
total
Example:
In a coin flip what is the probability of getting a heads?
• Total different outcomes – {H, T} = 2
• Total events – (H) = 1
1
• Thus P(H) = = 0.5 = 50%
2
Note: a probability can be expressed
as a fraction, decimal, or a percent.
Basic Probability Theory
success
P(E) 
total
Example:
In a die toss what is the probability of getting an even?
• Total different outcomes – {1,2,3,4,5,6} = 6
• Total events – (2,4,6) = 3
3
• Thus P(H) = = 0.5 = 50%
6
Note: a probability can be expressed
as a fraction, decimal, or a percent.
Basic Probability Theory
This can also be represented in a Venn diagram.
U
A
4
2
1
6
3
Set U = {all numbers on a die}
Set A = {even numbers on a die}
5
P(A) =
𝒏(𝑨)
𝒏(𝑼)
=
3
6
= 0.5
Basic Probability Theory
If all of the equally likely possible outcomes of a random
experiment can be listed as U, the universal set, and an
event A is defined and represented by a set A, then:
P(A) =
𝒏(𝑨)
𝒏(𝑼)
There are 3 consequences for this law:
𝒏(𝑼)
1. P(A) =
=1
The probability of a certain event is 1
𝒏(𝑼)
2. P(∅) =
𝒏(∅)
𝒏(𝑼)
=0
3. 0 ≤ P(A) ≤ 1
The probability of an impossible event is 0
The probability of an event always lies
between 0 and 1
Basic Probability Theory
Impossible and Certain
If P(E) = 0, then the event cannot occur.
It is impossible.
If P(E) = 1, then the event must occur.
It is certain.
0  P( E )  1
Basic Probability Theory
Practice
Find the probability that these events occur for the
random experiment ‘rolling a fair dice.’
a)
b)
c)
d)
Rolling an odd number
Rolling an even prime number
Rolling an odd prime number
Rolling a number that is either prime or even
Basic Probability Theory
Practice: A ticket is randomly selected from a
basket containing 3 green, 4 yellow and 5 blue
tickets. Determine the probability of getting:
a)
b)
c)
d)
a green ticket
a green or yellow ticket
an orange ticket
a green, yellow or blue ticket
Basic Probability Theory
Practice: A ordinary 6-sided die is rolled
once. Determine the chance of:
a)
b)
c)
d)
getting a 6
getting a 1 or 2
not getting a 6
not getting a 1 or 2
Basic Probability Theory
Practice: A bag has 20 coins numbered from 1 to 20.
A coin is drawn at random and its number is noted.
a) P (even) =
b) P (divisible by 3) =
a) P (divisible by 3 or 5) =
Basic Probability Theory
Practice: A family has three children.
a) List the sample space for the gender of the
children.
Find:
b) P (3 boys)
c) P (2 boys and 1 girl)
d) P (at least 2 girls)
Basic Probability Theory
Complementary Events
For complementary events:
P(A’) = 1 – P(A)
Example:
What is the probability of not randomly
drawing a queen in a deck of cards?
Solution:
Probability of drawing a queen is P(Q) =
Thus, not drawing a queen is P(Q’) = 1 –
4
52
4
52
=
48
52
=
12
13
Basic Probability Theory
Combined Events
For combined events:
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Example:
50 people are members at a gym. 28 of those people play
tennis. 14 of those people play basketball and 7 play both tennis
and basketball. What is the probability that a randomly selected
person plays tennis or basketball?
Solution:
28
14
• P(T) =
, P(B) =
, P(T ∩ B) =
•
28
50
+
50
14
50
-
7
50
=
50
35
50
=
7
10
7
50
Basic Probability Theory
Combined Events - Practice
Basic Probability Theory
Combined Events - Practice
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?
Basic Probability Theory
Combined Events - Practice
If P(A) = 0.6 and P(A  B) = 0.7 and P(A  B) = 0.3,
find P(B).
Conditional Probability
Let’s say there is a class:
In this class of 25 students,
With Venn Diagrams, we’ve:
16 study French, 11 study
But what if we addMalay
a condition?
and 4 study neither.
F
U
Been
to find the
What is the probability
thatable
a student
M
probability
chosen at random studies
French,of:
given
• A student studies F and M
studies
Malay?
10 that
6 the student
5
• A student studies only 1
4
•
•
language
A student studies no
language
A student doesn’t study F
Conditional Probability
What is the probability that a
student chosen at random
studies French, given that the
student studies Malay?
In this class of 25 students,
16 study French, 11 study
Malay and 4 study neither.
U
F
M
10
6
5
4
This is conditional probability
and is written P (F∣M)
Conditional Probability
Given that M has definitely
occurred we are restricted to
the set M (shaded) not the
universal set.
U
F
M
10
6
5
In this class of 25 students,
16 study French, 11 study
Malay and 4 study neither.
If we want to determine the
probability that F has also
occurred we are restricted to
the intersection of F and M
P (F∣M) =
4
𝒏 (𝑭∩𝑴)
𝒏(𝑴)
=
𝟔
𝟏𝟏
Conditional Probability
For conditional probability:
The conditional probability that A occurs given
that B has occurred is written as P (A∣B) and is
defined as:
P (A∣B) =
𝑷 (𝑨∩𝑩)
𝑷(𝑩)
Conditional Probability
Practice
In a class of 29 students, 20 study French, 15 study Malay,
and 8 students study both languages. A student is chosen from
random from the class. Find the probability that the student:
1.
2.
3.
4.
5.
6.
7.
Studies French
Studies neither language
Studies at least one language
Studies both languages
Studies Malay given that they study French
Studies French given that they study Malay
Studies both languages given that they study at
least one language.
Conditional Probability
Practice: 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?
Conditional Probability
Practice: 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
Conditional Probability
Practice: 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
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?
Mutually Exclusive Events
Two events, A and B are mutually exclusive if whenever A
occurs it is impossible for B to occur and vice-versa.
• Events such as A and A’ must be mutually exclusive.
• If events A and B are mutually exclusive, then A ∩ B = ∅
Events A and B are mutually exclusive
if and only if P(A ∩ B) = 0
Mutually Exclusive Events
Practice
The numbers 3, 4, 5, 6, 7, 8, 9 and 10 are each written on
on identical piece of card and placed in a bag. A random
experiment is: a random card is selected at random from
the bag.
Let A be the event ‘a prime number is chosen’ and B the
event ‘an even number is chosen.’
a) Draw a Venn diagram that describes the random
experiment.
b) Determine whether the events A and B are mutually
exclusive.
Mutually Exclusive Events
Please Note: If events are mutually exclusive, then
the probability of either one event or the other event
occurring is given by:
P( A  B)  P( A)  P( B)
P(either A or B) = P(A) + P(B)
Mutually Exclusive Events
Practice: 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
Independent Events
• Events where the occurrence of one of the
events ______
does _____
not affect the occurrence of
the other event.
A and B are independent if and only if:
P(A ∩ B) = P(A) × P(B)
P(A and B) = P(A) × P(B)
“and” → multiplication
P(A and B and C) = P(A) × P(B) × P(C)
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?
Independent Events
Practice
The numbers 2, 3, 4, 5, 6, 7, 8 and 9 are each written on on
identical piece of card and placed in a bag. A card is
selected from random from the bag.
Let A be the event ‘an odd number is chosen’ and B the
event ‘a square number is chosen.’
a) Draw a Venn diagram to represent the experiment.
b) Determine whether the events A and B are independent.
Independent Events
Practice: A coin and a die are tossed simultaneously.
Determine the probability of getting heads and a 3.
Independent Events
Practice: There are 9 brown boxes and 6 red boxes
on a shelf. Anna chooses a box and replaces it.
Brian does the same thing. What is the probability
that Anna and Brian choose a brown box?
Independent Events
Practice
P (B) = 1/3
and P(A  B) = p
P (A) = ½
Find p if:
a) A and B are mutually exclusive
b) A and B are independent
Dependent Events
There are 9 brown boxes and 6 red boxes on a shelf.
What if Anna choose the box and did not replace it?
Then Brian’s event of choosing a box becomes dependent.
If Anna chooses red,
If Anna chooses brown,
P(Brian chooses brown) =
9
14
P(Brian chooses brown) =
8
14
P(Anna then Brian choose brown)

9 8

15 14

12
35
Dependent Events
• Events where the occurrence of one of the
does affect the occurrence of the
events ______
other event.
P(A then B) = P(A) × P(B given that A has occurred)
Dependent Events
Practice: A box contains 4 red and 2 yellow tickets.
Two tickets are randomly selected one by one from the
box, without replacement. Find the probability that:
(a) both are red
(b) the first is red and the second is yellow.
Dependent Events
Practice: A hat contains tickets with numbers 1, 2, 3, … , 19, 20
printed on them. If 3 tickets are draw from the hat, without
replacement, determine the probability that all are prime numbers.
Basic Probability Theory
Experimental Probability
• One way to determine probability is to do an
experiment and analyze the results.
• Trials = the total number of times the
experiment is repeated.
• Outcomes = the different results possible
for one trial of the experiment
Basic Probability Theory
Experiment
1
Experimental
 Theoretical
• Toss a coin 10 times
performed
inof
antimes
experiment,
then
• InThe
eachmore
case,trials
record
the number
you get Heads
the experimental
probability will approximately
• Place
results on the board
equal the theoretical probability.
Experiment 2
• Roll a die 20 times
• In each case, record the number of times you get 5
• Place results on the board
Sample Space
A sample space is the set of all
possible outcomes of an experiment.
4 ways to create a Sample Set:
i.
List the outcomes {you must use curly brackets}
ii. Table of outcomes
iii. Draw a 2 Dimensional Grid
iv. Draw a Tree Diagram
Sample Space
Practice
List the outcomes for:
a) Flipping a coin
b) Rolling a single die
c) Flipping two coins
d) Rolling two dice
Sample Space
Practice
Use a 2D grid to illustrate the sample space for tossing a coin
and rolling a die simultaneously.
Find the probability of:
Rolling a Die
Coin
Toss
1
2
3
4
5
6
H
H1
H2
H3
H4
H5
H6
T
T1
T2
T3
T4
T5
T6
a) tossing heads on the coin
b) getting tails and a 5
c) getting tails or a 5
Sample Space
Practice
Two square spinners, each with 1, 2, 3, and 4 on their edges, are
twirled simultaneously. Draw a 2D grid of the possible outcomes.
Find the probability of:
Spin 2
Spin 1
1
2
3
4
1
(1,1)
(1,2)
(1,3)
(1,4)
2
(2,1)
(2,2)
(2,3)
(2,4)
3
(3,1)
(3,2)
(3,3)
(3,4)
4
(4,1)
(4,2)
(4,3)
(4,4)
a) getting a 3 with each spinner
b) getting a 3 and a 1
c) getting an even result for each spinner
Sample Space
Practice: A red and blue dice are rolled together. Calculate the
probability that: (a) The total score is 7 (b) The same number
comes up on both dice (c) The difference between the scores is
1 (d) The score on the red dice is less than the score on the
blue dice (e) The total score is a prime number
Roll 2
Roll 1
1
2
3
4
5
6
1
(1,1) 2
(1,2) 3
(1,3) 4
(1,4) 5
(1,5) 6
(1,6) 7
2
(2,1) 3
(2,2) 4
(2,3) 5
(2,4) 6
(2,5) 7
(2,6) 8
3
(3,1) 4
(3,2) 5
(3,3) 6
(3,4) 7
(3,5) 8
(3,6) 9
4
(4,1) 5
(4,2) 6
(4,3) 7
(4,4) 8
(4,5) 9
(4,6) 10
5
(5,1) 6
(5,2) 7
(5,3) 8
(5,4) 9
(5,5) 10 (5,6) 11
6
(6,1) 7
(6,2) 8
(6,3) 9
(6,4) 10 (6,5) 11 (6,6) 12
Sample Space
Practice
Draw a table of outcomes to display the possible results
when two dice are rolled and the scores are summed.
Determine the probability that the sum of the dice is 7.
Roll 2
Roll 1
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
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)
Tree Diagrams
Consider
3
of
4
In archery, Li has a probability
hitting a target and Yuka
has a probability of 4 . They both shoot simultaneously.
5
Yuka
4
5
Li
3
4
1
4
H
1
5
4
5
H
M
H
Outcome
Probability
H and H
3 4

4 5
𝟏𝟐
𝟐𝟎
H and M
3 1

4 5
𝟑
𝟐𝟎
M and H
1 4

4 5
𝟒
𝟐𝟎
M and M
1 1

4 5
M
1
5
M
𝟏
𝟐𝟎
total = 20/20
Tree Diagrams
Bike
Car
8
10
2
10
6
10
S
N
4
10
6
10
4
10
S
SS
48
100
N
SN
32
100
S
NS
N
NN
Tree Diagrams
7
12
Consider a box containing:
3 red, 2 blue, 1 yellow marble
With Replacement
9
36
Without Replacement
6
30
What’s the probability of getting two red marbles?
Tree Diagrams
9/36
6/36
6/36 + 3/36 + 6/36 + 2/36 + 3/36 + 2/36
= 22/36
3/36
6/36
4/36
2/36
3/36
2/36
1/36
What’s the answer for b?
Tree Diagrams
50
56