Transcript Probability - mslaurenphillips

PROBABILITY
Questions, comments, concerns?
Ok to move on?
Vocab
 Trial- number of times an experiment is
repeated
 Outcomes- different results possible
 Frequency- number of times an event occurs
 Relative Frequency- ??
Warm Up
List all the possible outcomes for tossing a coin
once…
Head, Tail
or
HT
Let an event be tossing a single coin two times. Represent the
sample space :
a. In a list form
b. in a table form
c. as a tree diagram.
HH, HT,
TH, TT
Toss 1 Toss 2
H
H
H
T
T
H
T
T
Practice
An urn contains 5 purple, 3 white, and 4 red
marbles. Two marbles are drawn. Draw a
sample space for the event using a TREE
DIAGRAM.
Practice
 Let the event be a couple having children.
Draw a tree diagram for the outcomes of a
couple having four children.
Practice
Let the event be tossing a coin and then rolling
a fair six-sided die. Draw a tree diagram and list
the outcomes
Practice
 The town of Alright has a population of 200
adults: 60% are women and 10% of them are
more than 6ft tall. Of the men, 30% are more
than 6 ft tall. Draw and label a tree diagram
of the town.
NOW the juicy stuff…
Probability!
Probability:
Formula in set notation?
Theoretical vs Experimental Probability?
Practice
Find the probability that a 7 will show when
rolling a fair six-sided die whose faces are
numbered 1 through 6.
Practice
 Find the probability that an even number
shows when a six-sided die is rolled.
Review
A compliment is….
P(A’) = 1- P(A)
Ex: If there is a 35% chance it will
rain tomorrow, what is the probability
it will NOT rain.
Practice
Find the probability that when rolling two dice
they will not show doubles.
Laws of Probability
Mutually Exclusive: P (A U B) = P(A) + P(B)
Combined Events: P(A U B) = P(A) + P(B) – P(A∩ B)
Independent Event: P(A ∩ B) = P(A) ∙ P(B)
𝑃(𝐴∩𝐵)
Dependent Event: P (B|A) =
𝑃(𝐴)
Note: OR = Union
AND = Intersection
Mutually Exclusive
If it is not possible for the events to occur at the
same time.
Mutually Exclusive
Probability
 P(A U B) = P(A) + P(B)
 where sets A and B have no
elements in common.
Example
Let A be the event that a person is a female and
B be the event that a person is 1.6 metres tall.
a. Are the two mutually exclusive?
b. Justify your answer by drawing a venn
diagram
Food for thought
Can two political parties be considered mutually
exclusive?
Practice
Find the probability of having a 5 or an even
number show when rolling a six-sided die.
Combined Events
 These events are NOT mutually exclusive.
Combined Events Probability
P(A U B) = P(A) + P(B) – P(A ∩ B)
Consider the following: U= numbers 1-12. A
= multiples of 3 numbers. B= Even numbers.
Create a Venn diagram.
What is P(A U B)?
Practice
 Find the probability, when drawing a single
card from a standard 52-card deck, that a
card is either a nine or a face card.
Practice
 Find the probability, when drawing a single
card from a 52-card deck, that a card is either
red or an ace.
Practice
 If P(A) = .37, P(B) = .55, P(A∩B) = .13, find:
 P(AUB)
 P(A’)
 P(B’)
 P(A∩B’)
Data was collected for which gender reads which
types of book. A book as selected at random. Find
the probability that:
a. The book was read by a female or it was
an action book.
b. The book was a historical book or it was
read by a male.
c. The book was not read by a male and it
was an action book.
Action
Romance
Historical
Male
10
17
8
Female
3
4
8
Independent Events
This occurs if the probability of the first event
does NOT effect the probability of the second
event…
Then to see what the probability of both
happening is
P(A∩B) = P (A) ∙ P(B)
Practice
If the probability it will snow on Monday is 25%
and on Tuesday it is 40%, what is the probability
that it will snow on both Monday and Tuesday?
Practice
Find the probability of the toss of a coin landing
heads up and the roll of a die showing a four.
Practice
 An urn contains 4 red marbles and 3 green
marbles. Find the probability of drawing a red
marble, replacing it, drawing a green marble,
replacing it, and then drawing a red marble.
From a group of 20 athletes it is found that 13 played
billiards, 12 played golf and 5 played both billiards
and golf.
a. Draw a venn diagram to represent this
information.
b. Find the probability that an athlete chosen at
random
a.
b.
c.
d.
Plays golf
Does not play billiards
Plays billiards and golf
Plays billiards or golf
c. State a reason as to whether or not the events
are independent.
d. State a reason as to whether or not the events
are mutually exclusive.
Dependent Events
 Two events are said to be dependent if the
probability of the first event occurring
influences the probability of the second event
occuring.
Example
An urn contains 4 red marbles and 3 green
marbles. Find the probability of drawing a red
marble, NOT replacing it, and then drawing a
green marble.
Practice
When drawing two cards from a standard 52card deck, find the probability of both cards
being black.
A bag contains two red sweets and three green
sweets. Jacques takes one sweet from the bag,
notes its colour, then eats it. He then takes
another sweet from the bag. Copy and complete
the tree diagram below to show all possibilities.
Conditional Probability
Recognizable by the words “given that”
𝑃(𝐴∩𝐵)
P(B|A) =
𝑃(𝐴)
Practice
 Six identical marbles, numbered, 1,2,3,4,5,6
are placed in an urn. A single draw is made. It
is known only that an even-numbered marble
is selected. What is the probability that it is a
four?
Practice
Consider the data collected:
Sports Car Truck
Sedan
Totals
Junior
8
7
9
24
Senior
15
4
7
26
Totals
23
11
16
50
a. Find the probability that the kind of vehicle
driven to school was a truck, given that the
student was a senior.
b. Given that the vehicle driven to school was a
sports car, find the probability that the
student was a junior.
Practice
From a standard 52 card deck, let event D be the
selection of a diamond and event F be the selection
of a face card. One card is selected at random. Find
the probability that:
a. P(the card is a diamond or a face card)
b. P(the card is neither a diamond nor a face
card)
c. P(a face card is selected given that a
diamond has already been selected.)
d. P(a diamond is selected given that a face
card has already been selected. )