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 experimentEvents: subsets of the outcomes of an

experimentEmpirical probability:

The number of times an event is an outcomeP(E) = The number of times the experiment is run

This is used when the theoretical probability cannot becalculated (we'll talk about that shortly)

Empirical probability

The number of times an event is an outcomeP(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

ExampleA coin is tossed 50 times and it comes up heads 29 of those times. What is empirical probability of flipping a head?

29P(head) = = 0.5850

What is the probability of flipping a tail?Tail = 50 - 29 = 21

P 21(tail) = = 0.4250

Note that P(head) + P(tail) = 1 0.58 + 0.42 = 1

AnotherStudents at KU were polled about their favorite search engines. Here are the results.Google 63Dogpile 29Yahoo 34Bing 24If one KU student was randomly chosen to doa 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

63P(google) = = 0.42150

29P(dogpile) = = 0.19 (rounded)150

34P(yahoo) = = 0.23 (rounded)150

24P(bing) = = 0.16150

Note that the sum of these is 1:0.42 + 0.19 + 0.23 + 0.16 = 1

A few more definitionsEqually likely outcomes: each of the outcomes of an

experiment has the same chance of occurringTheoretical probability (equally likely outcomes):

Law of large numbers: When the number of ‘experiments is very large the empirical probability is the same as the theoretical probability.

number of outcomes favorable to EP(E) = Total number of possible outcomes

Note that this is different from empirical probablity because it does not require an experiment.

Most common example Dice

Dice have 6 possible outcomes: 1,2,3,4,5,6Total number of outcomes = 6

What is the probability of rolling a 4?1P(4) = 6

What is the probablity of rolling an odd number?(1,3,5) there are 3 odd numbers

P 3 1(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(occ

ur) = 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 1P(ace) = =

52 13What is the probability of not selecting an ace?

1P (not ace) = 1 - 13

13 1 12P (not ace) = - 13 13 13

What is the probability of selecting a heart?13 1P(heart) = = 52 4

What is the probability of selecting a number card?Total # = 4(2 through 10) = 4(9) = 36

36 9P(number) = = 52 13

What is the probability of selecting a card between 5 and 9?Total = 4(6 through 8) = 4(3) = 12

12 3P(5-9) = = 52 13

Odds against

The odds against an event occuring is the probablitythat the event will not occur divided by the probability that the event will occur

P(not occur) failureOdds against = = P(occur) success

ExampleWhat is the odds against drawing a queen from a deck of card?

4 1P(queen) = = 52 13

1P(not queen) = 1 - 13

13 1 12P(not queen) = - = 13 13 13

12P(not queen) 13Odds against = = note this is divis1P(queen)

13

ion:

12 1Odds against = 13 1312Odds against = 13

13 1

12Odds against = 1

Odds in favorP(event) successOdds in favor = =

P(not event) failureWhat is the odds in favor of drawing a queen from a deck of card?

4 1P(queen) = = 52 13

1P(not queen) = 1 - 13

13 1 12P(not queen) = - = 13 13 13

Odds in

11 1213favor = 12 13 13

131Odds in favor =

13

13 1 = 12 12

1Note: Odds in favor = Odds against

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) 7Odd against = =P(cat) 2

P(cat) = 2Total odds = 7 + 2 = 9

2P(cat) = 9

Expected value: used to determine probability over the long term (investments etc.)

E = P A Where P is the probability of an event occuring and A is the net amount or

i

1 1 2 2

loss if that event occurs.E = P A + P A .......P A

n 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 2prize is and the consolation prize is 100 100

97and for winning no prize is 100

1 2 97E = (48) + (18) + ( 2)100 100 10048 36 194E = + -

100 100 100 48 + 36 E =

- 194100

110E = - = -1.10100

The expected value of each ticket is - $1.10

Fair price = expected value – cost to playthis is the ‘break even’ price

For the previous example:expected value = -1.10Cost to play = 2.00Fair price = -1.10 + 2.00 = 0.90Each ticket should cost $.72 for the expected value to be zeroNote:

1 2 9E = (50 .90) + (20 .90) + 100 100

7 ( .90)

10049.1 38.2 87.3E =

100E = 0

Tree diagramsCounting 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 ballsof the same color

3 1P(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 successfuloutcome is the sum probability of the first event and secondevent 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 theproduct 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 3Let B be the probability of an even faced diceP (A or B) = P(A) +P(B) - P(A + B) (addition formula)

1P(A) = 63 1P(B) = = (2,4,6 are the possible dice faces)6 2

P(A

+ B) = 0 (you cannot have a 3 that is even)1 1P(A or B) = + - 06 21 3P(A or B) = + 6 64 2P(A or B) = 6 3

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 ball2 1P(A) = 6 31P(B) = 5

P(A + B) = P(A) P(B)1 1 1P(A + B) = 5 3 15

Thank you for attending!