Transcript Welcome to our seventh seminar!

Welcome to our seventh
seminar!
We’ll begin shortly
Definitions
Experiment: an act or operation for the purpose
of discovering something unknown.
Outcome: the results of an experiment
Events: subsets of the outcomes of an
experiment
Empirical probability:
The number of times an event is an outcome
P(E) =
The number of times the experiment is run
This is used when the theoretical probability cannot be
calculated (we'll talk about that shortly)
Empirical probability
The number of times an event is an outcome
P(E) =
The number of times the experiment is run
Since the top number is always either smaller than
or equal bottom number, P(E) will always be a
number between 0 and 1.
It may be expressed as a fraction or a decimal
Example
A coin is tossed 50 times and it comes up heads 29
of those times. What is empirical probability of
flipping a head?
29
P(head) =
= 0.58
50
What is the probability of flipping a tail?
Tail = 50 - 29 = 21
P(tail) =
21
= 0.42
50
Note that P(head) + P(tail) = 1
0.58 + 0.42 = 1
Another
Students at KU were polled about their
favorite search engines.
Here are the results.
Google 63
Dogpile 29
Yahoo
34
Bing
24
If one KU student was randomly chosen to do
a search on the internet, what is the probablity
that the student will choose:
Google? Dogpile? Yahoo? Bing?
(continued..)
First find the total number of "experiments"
Total = 63 + 29 + 34 + 24 = 150
63
P(google) =
= 0.42
150
29
P(dogpile) =
= 0.19 (rounded)
150
34
P(yahoo) =
= 0.23 (rounded)
150
24
P(bing) =
= 0.16
150
Note that the sum of these is 1:
0.42 + 0.19 + 0.23 + 0.16 = 1
A few more definitions
Equally likely outcomes: each of the outcomes of an
experiment has the same chance of occurring
Theoretical probability (equally likely outcomes):
number of outcomes favorable to E
Total number of possible outcomes
Note that this is different from empirical
P(E) =
probablity because it does not require an
experiment.
Law of large numbers: When the number of
‘experiments is very large the empirical
probability is the same as the theoretical
probability.
Most common example Dice
Dice have 6 possible outcomes: 1,2,3,4,5,6
Total number of outcomes = 6
What is the probability of rolling a 4?
1
P(4) =
6
What is the probablity of rolling an odd number?
(1,3,5) there are 3 odd numbers
3
1
P(odd) = =
6
2
Hints
• If an event cannot occur P(E) = 0 (such as
rolling a 9 on the dice)
• If a probability must occur P(E) = 1 (such as
flipping a double headed coin)
• 0 ≤ P(E) ≤ 1

 P(E) = 1

The sum of all event probablities is 1
The probability that an event will occur
and not occur is 1
P(occur) + P(not occur) = 1
so: 1 - P(occur) = P(not occur)
Example (deck of cards)
Data for a deck of cards:
Total cards: 52
# 7's (or any number) = 4
# hearts, clubs, spades, diamonds = 13
# jacks, queens, kings, aces = 4
What is the probablity of selecting an ace?
4
1
P(ace) =
=
52 13
What is the probability of not selecting an ace?
1
P (not ace) = 1 13
13 1
12
P (not ace) =

13 13 13
What is the probability of selecting a heart?
13
1
P(heart) =
=
52
4
What is the probability of selecting a number card?
Total # = 4(2 through 10) = 4(9) = 36
36
9
P(number) =
=
52 13
What is the probability of selecting a card between 5 and 9?
Total = 4(6 through 8) = 4(3) = 12
P(5-9) =
12
3
=
52 13
Odds against
The odds against an event occuring is the probablity
that the event will not occur divided by the probability
that the event will occur
P(not occur)
failure
Odds against =
=
P(occur)
success
Example
What is the odds against drawing a queen from a deck of card?
4
1
P(queen) =
=
52 13
1
P(not queen) = 1 13
13 1
12
P(not queen) =
=
13 13 13
Odds against =
Odds against =
Odds against =
Odds against =
12
P(not queen)
= 13 note this is division:
1
P(queen)
13
12
1

13 13
12
13

13
1
12
1
Odds in favor
P(event)
success
=
P(not event)
failure
What is the odds in favor of drawing a queen from a deck of card?
4
1
P(queen) =
=
52 13
1
P(not queen) = 1 13
13 1
12
P(not queen) =
=
13 13 13
1
1
12
Odds in favor = 13 

12
13 13
13
1
13
1
Odds in favor =

=
13 12
12
1
Note: Odds in favor =
Odds against
Odds in favor =
Example
If the odds against of a cat being a patient
at a vets office is 7 to 2, what is the probability?
P(not cat) 7
Odd against =
=
P(cat)
2
P(cat) = 2
Total odds = 7 + 2 = 9
2
P(cat) =
9
Expected value: used to determine probability
over the long term (investments etc.)
E =  Pi A
Where P is the probability
of an event occuring and
A is the net amount or loss
if that event occurs.
E = P1A1 + P2 A 2 .......Pn A n
One hundred raffle tickets are sold for 2$ each. The grand prize is
50$ and two 20$ prizes are consolation prizes . What is the
expected gain?
Probability for each ticket winning the grand
1
2
prize is
and the consolation prize is
100
100
97
and for winning no prize is
100
1
2
97
E=
(48) +
(18) +
( 2)
100
100
100
48
36 194
E=
+
100 100 100
48 + 36 - 194
E=
100
110
E== -1.10
100
The expected value of each ticket is - $1.10
Fair price = expected value – cost to play
this is the ‘break even’ price
For the previous example:
expected value = -1.10
Cost to play = 2.00
Fair price = -1.10 + 2.00 = 0.90
Each ticket should cost $.72 for the expected value to be zero
Note:
1
2
97
E=
(50  .90) +
(20  .90) +
(.90)
100
100
100
49.1  38.2  87.3
E=
100
E=0
Tree diagrams
Counting principle: If the first experiment can be done M ways
and a second can be done N ways, then the two experiments
MN can be done M*N ways.
Barney has three pairs of jeans and three shirts to choose from.
M = 3, N = 3 so MN = 3*3 = 9
There are 9 possible outcomes.
If we have a box with two red, two green and two white
balls in it, and we choose two balls without looking,
what is the probability of getting two balls of the same
color?
There are 9 possible outcomes..
Note that 3 of these outcomes are balls
of the same color
3 1
P(RR,WW,GG) = =
9 3
Or and And problems
“Or” problems have a successful outcome for at
least one of the events
“And” problems have a favorable outcome for
each of the events
P (A or B) = P(A) +P(B) - P(A + B)
(addition formula)
The probability that there will be at least on successful
outcome is the sum probability of the first event and second
event occurring minus the probablity that both will occur
P(A and B) = P(A)  P(B)
The probability that all outcomes with be favorable is the
product of probablities of each event occurring.
If I roll a dice, what is the probability that the
outcome will be 3 or an even number?
Let A be the probability of a 3
Let B be the probability of an even faced dice
P (A or B) = P(A) +P(B) - P(A + B)
(addition formula)
1
P(A) =
6
3
1
P(B) = =
(2,4,6 are the possible dice faces)
6
2
P(A + B) = 0 (you cannot have a 3 that is even)
1 1
+ -0
6 2
1
3
P(A or B) = +
6
6
4 2
P(A or B) = 
6 3
P(A or B) =
If we have a box with two red, two green and two white
balls in it, and we select two balls one at a time what is
the probability that the first ball will be red and then
the second ball will be red?
Let A be the first ball and B be the second ball
2 1
P(A) = 
6 3
1
P(B) =
5
P(A + B) = P(A)  P(B)
1 1 1
P(A + B) =  
5 3 15
Thank you for attending!