Download Probability Notes (Summary PPT)

You probability
wonder what we’re
going to do next!
Probability Basics

Experiment


Sample space


An activity with observable results or
outcomes
The set of all possible outcomes for an
experiment
Event

Any subset of the sample space
Probability Basics —
General Definition
n(E)
P(E) 
n(S)
where P(E) represents the probability of an
event E occurring, n(E) represents the
number of individual outcomes in the event E,
and n(S) represents the number of individual
outcomes in the sample space S.
Flip a coin



A well-known statistician named Karl
Pearson once flipped a coin 24,000 times
and recorded ________ “heads”; this
result is extremely close to the theoretical
expected value!
P(H) = _____
P(T) = _____
Expected # of H = P(H) x 24,000 = _____
Spinners
Spin each spinner once. Find the probability
that the spinner lands in region A.
A
A
A
C
B
B
D
C
Spinners
If S = {1, 2, 3, 4, 5, . . . , 22, 23, 24}, find
the probability of the spinning of a
 Prime number
 Even number
 Number less than 10
 Number less than 3 or greater than 17
 Number less than 12 and greater than 9
Rolling Dice
Roll a single die once. Find the following
probabilities:
P(number greater than 4 or less than 2)
 P(odd or even number)
 P(number greater than 10)
 P(at least 3)

Probability Vocabulary

Complementary event
 Everything else (besides the outcomes
in the event) in the sample space
 Examples:
 If A = “roll a 1 or a 2 on a die”, then
“A complement” = “roll a 3, 4, 5, or 6
on a die”.
 If R = “it rains today”, then R
complement = “it doesn’t rain today”.
Standard Cards

Find the probability of drawing
an ace from a standard deck of
playing cards.

Find P(“face card”)

Find P(card with a value between 4 and 9)
More Vocabulary

Mutually exclusive events (Disjoint sets):
When one event occurs, the other
cannot possibly occur; the events have
no overlap
 Example:
 If A = “roll an even number” and B =
“roll a 3 or a 5”, find P(A or B) and
find P(A and B).

Probability of A or B

Mutually exclusive events
n(A)  n(B)
P(A or B)  P(A  B) 
n(S)

Non-mutually exclusive events
n(A)  n(B)  n(A  B)
P(A  B) 
n(S)
Probability of A or B


Draw a card out of a standard 52-card
deck. Find the probability that the card is
either: (a) a black card or an ace (b) a
red card or a club
Roll a die once. If A = “roll an even
number” and B = “roll a 5 or a 6”,
find P(A or B).
Fundamental Counting Principle

If event M can occur in m ways and after
it has occurred, event N can occur in n
ways, then event M followed by event N
can occur in m x n ways.
(P.S. A tree diagram helps!)
Fundamental Counting Principle



How many outcomes are there for flipping
3 coins?
How many outcomes are there for rolling
2 dice?
If I have 6 pairs of pants and 8 shirts from
which to choose, how many outfits can I
pick?
Fundamental Counting Principle

If automobile license plates consist of 4
letters followed by 3 digits (and repetition
of letters and digits is allowed), how many
different license plates are possible?
(This time, a tree diagram isn’t
encouraged.)
Multi-stage Experiments

For any multi-stage experiment, the
probability of the outcome along any
path of the tree diagram is equal to
the product of the probabilities along
the path.
Toss 2 coins


List the sample space. Use set notation
and a tree diagram.
Find the probability of
at least one head.
The

Problem
If the chance for success on the first stage
of a rocket firing procedure is 96%, the
chance for success on the second stage is
98%, and the chance for success on the
final stage is 99%, find the probability for
success on all 3 stages of the rocket firing
procedure.
Rolling Two Dice: Sample Space
Rolling Two Dice
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
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Find the probability of a 3 on the first roll
and a 3 on the second roll of a die.
Find the probability of a sum of 7.
Find the probability of a sum of 10 or
more.
Find the probability that both numbers are
even.
Independent Events


When the outcome of one event has no
influence on the outcome of a second
event, the events are independent.
For any independent events A and B,
P(A and B) = P(A) x P(B).
Draw a ball from a container,
replace it, and then draw a 2nd ball.

Find the probability of a red, then a red.

Find P(no ball is red).

Find P(at least one red).

Find P(same color).
Draw a ball from a container, don’t
replace it, and then draw a 2nd ball.
(dependent events)

Find P(red, then green).

Find P(no ball is red).

Find P(same color ball).
A bag contains the letters of the
word “probability”.

Draw 4 letters, one by one, from the bag.
Find the probability of picking the letters
of the word “baby” if the letters are drawn

With replacement

Without replacement
Geometric Probabilities

If a dart hits the target below, find the
probability that it hits somewhere in
region 1.
2
1
3
4
The radius of
the inner
circle is 1
unit, and the
radius of the
outer circle is
2 units.
1
2
For a challenge, or two, or three!

“Pascal’s Probabilities”

“The Prisoner Problem”

“The Birthday Problem”
Using Simulations

Flipping a coin

Rolling a die

Find the probability of a married couple
having 2 boys and 2 girls.
Isn’t
that
odd?
P(for)
Odds for an event 
P(against)
P(A)
P(A)
Odds for an event 

P(A) 1 - P(A)
Odds



Find the odds for tossing a “head” on a
fair coin.
Find the odds for rolling a sum of 7 on the
roll of two dice.
Find the odds for drawing a card valued
from 1 (ace) to 8, inclusive, from a
standard 52-card deck.
Conditional Probabilities

When the sample space of an experiment
is affected by additional information
PA  B
P(B given A)  P(B | A) 
PA 
Conditional Probabilities


If A = “getting a tail on the 1st toss of a
coin” and B = “getting a tail on all three
tosses of a coin”, find P(B|A).
What is the probability of rolling a 6 on a
fair die if you know that you rolled an
even number?
Expected Value

If, in an experiment, the possible
outcomes are numbers a1, a2, a3, . . . , an
occurring with probabilities p1, p2, p3, . . . ,
pn, respectively, then the expected value,
E, is given by the equation
E = a1 p1 + a2 p2 + a3 p3 +
...,
+ an pn.
Expected Value (level 1)


Flip a coin 1,000 times. How many heads
do you expect?
Roll a pair of dice 60 times. How many
times do you expect a sum of 5?
Expected Value (level 2)


If a player gets $2 if the
A
spinner lands on A, $4 for
landing on B, $4 for C, and
C
$1 for D, what is the
expected payoff for this game?
B
A
D
If the game costs $3 to play, is this a fair
game?
Factorial Notation

0! = 1 (by definition)

Compute:
5!
7!
3!
6!
4! 2!
Permutations

From n objects, choose r of them and
arrange them in a definite order. The
number of ways this can be done is given
by
n!
P

n r
n  r !
Permutations (Correspondences)

How many different
ways can 4 swimmers
(Al, Betty, Carol, and Dan)
be placed in 4 lanes for a
swim meet?
Permutations

If there are 12 players on a little league
baseball team, how many ways can the
coach arrange batting orders, with 9
positions in the field
and at bat?
Combinations

From n objects, choose subsets of size r
(order is unimportant). The number of
ways this can be done is given by
P
n!
n r
C


n r
r! r!n  r !
Combinations
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
With 9 club members, how many
different committees of 4 can be selected
to attend a conference?
Braille Activity
Permutations & Combinations


How many games are played in a women’s
soccer conference if there are 8 teams and
all teams play one another once?
There are 10 members of a club. How
many different “slates” could the
membership elect as president, vicepresident, and secretary/treasurer (3
offices)?
Probability (with
permutations & combinations)

Given a class of 12 girls and 9 boys,
 In how many ways can a committee of
5 be chosen?
 In how many ways can a committee of
3 girls and 2 boys be chosen?
 What is the probability that a committee
of 5, chosen at random, consists of 3
girls and 2 boys?