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probability
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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
n(E)
P(E) =
(classical) or
n(S)
f
(experimental)
n
where P(E) means the probability of an event
occurring, n(E) means the number of individual
outcomes in the event, and n(S) means the
number of individual outcomes in the sample
space.
Flip a coin


A well-known statistician named Karl
Pearson once flipped a coin 24,000 times
and recorded _______ “heads”; this
result was extremely close to the
theoretical probability and expected
number of heads.
P(H) = _____ E(H) = _________
Spinners
A
D
B
A
B
C
Spin each spinner once. Find P(A).
C
Spinners

If S = {1, 2, 3, 4, 5, . . ., 22, 23, 24},
find the probability 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
Dice

Roll a single die once. Find the
following probabilities:




P(number greater than 4 or less than 2)
P(odd or even)
P(greater than 10)
P(at least 3)
Odds

Odds for an event =
P(for an event)/P(against the event)
If the odds for a horse winning a race are
given, find the probability that the horse
wins the race.
(a) 2:5
(b) 5:1
(c) 1:30
Experimental Probabilities

Conduct a poll to determine the
probability:



That a Gordon transfer student drives a
truck.
That a Gordon student has enjoyed a meal
at the Dining Hall.
That a Gordon student has visited the
BCM.
More vocabulary

Complementary events

Everything else in the sample space
 Examples:
 If A = rolling a 1 or a 2 on a die,
“A complement” is rolling
a 3, 4, 5, or 6
 If R = it rains today,
R = it
doesn’t rain today
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 (disjoint) events

When one event occurs, the other
cannot possibly occur
 Examples:
 If A = even # on the roll of a die
and B = 3 or 5,
P(A and B)  0
P(A or B)

Mutually exclusive events
P(A  B) 

n A   n B
= P(A) + P(B)
n(S)
Non-mutually exclusive events
n A  n B  n A  B
P A  B 
= P(A) + P(B) – P A  B
n S
P(A or B)


Draw a card out of a standard 52-card
deck. Find the probability that the
card is either black or an ace.
Roll a die once.
If A = “even number on the die”
and B = “rolling a 5 or 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.
(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? 3 dice?
If an ice cream shop has 32 flavors from
which to choose and 7 toppings, how many
different possibilities can I choose?
Fundamental Counting Principle

If automobile license plates consist of
4 letters followed by 3 digits, how many
different license plates are possible if
letters and digits may be repeated?
Multi-Stage Experiments

For any multi-stage experiment, the
probability of the outcome along any
path of a tree diagram is equal to the
product of the probabilities along the
path.
Problem

If the chance for success on the first
stage of a rocket firing procedure is 96%,
the second stage is 98%, and the final
stage is 99%, find the probability of
success on all three stages of the rocket
firing procedure.
Flipping coins




List the sample space for two coins.
(Use a tree diagram.)
Find the probability of at least one
head.
List the sample space for three
coins. (Use a tree diagram.)
Find the probability of exactly two
heads.
Rolling Two Dice





List the sample space.
Find the probability of a 3 on the first
and a 3 on the second.
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.
Rolling Two Dice
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 second 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).
Dependent events

When the outcome of one event has an
influence on the outcome of a second
event, the events are dependent.
Draw a ball from a container, don’t
replace it, and then draw a second ball.

Find the probability of a red, then a
blue.

Find P(no ball is red).

Find P(same color).
Dependent vs. Independent events
Consider a bag that contains 219 coins of which 6 are rare Indian
pennies. For the given pair of events A and​ B, complete parts​ (a) and​
(b) below.
A: When one of the coins is randomly​ selected, it is one of the Indian pennies.
B: When another one of the coins is randomly​ selected, it is also one of the
Indian pennies.


a. Determine whether events A and B are independent or
dependent.​ (If two events are technically dependent but can be
treated as if they are independent according to the​ 5% guideline,
consider them to be​ independent.)
b. Find​ P(A and​ B), the probability that events A and B both occur.
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
Consider the following 2 containers.

MATH
HAT
#1
#2
If a container above is selected at random,
and then a letter is selected at random from
the chosen container, what is the probability
that the letter chosen is an M?
For a challenge!

The Prisoner Problem

The Birthday Problem
Geometric Probabilities

If a dart hits the target below, find the
probability that it hits somewhere in
region 1.
2
2
1
1
2
3
4
The radius of the inner circle is 2
units and the radius of the outer
circle is 4 units.
Games


A game is played by drawing 3 cards
and replacing the card each time. A
player wins if at least one face card is
drawn. Find the probability of winning
this game.
Find the probability of a “Yahtzee” in
one throw of 5 dice.
Using Simulations

Flipping a coin

Rolling a die

Find the probability of a married couple
having 2 baby girls.
Conditional Probabilities

When the sample space of an experiment
is affected by additional info
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 3
tosses of a coin”, find P(B|A).
What is the probability of rolling a 6 on a fair
die if you know you rolled an even number?
Conditional Probabilities

With an auto insurance company, 60% of its
customers are considered low-risk, 30% are
medium-risk, and 10% are high-risk. After a
study, the company finds that during a 1-yr
period, 1% of the low-risk drivers had an
accident, 5% of the medium risk drivers had
an accident, and 9% of the high risk drivers
had an accident. If a driver is selected at
random, find the probability that the driver
will have had an accident during the year.
Factorial Notation

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
n
Pr 
n!
n  r !
Correspondences

How many different
ways can 4 swimmers
(Al, Betty, Carole, 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 on the field and at bat?
Permutations

How many arrangements are possible
with the letters in the word:

algebra?

statistics?
Combinations

From n objects, choose subsets of size r
(order unimportant). The number of
ways this can be done is
n
Cr 
n Pr
r!

n!
r ! n  r !
Combinations


With 9 club members, how many
different committees of 4 can be
selected to attend a conference?
Braille Activity
Permutations & Combinations


There are 10 players on a U-6 soccer team,
and the coach picks 5 starters. How many
different groups of starters can the coach
choose?
There are 10 members of a club. How many
different “slates” could the membership elect
as president, vice-president, 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?