ProbabilityBasic Concepts

Experiment

•An activity or measurement that results in an outcome we cannot predict with absolute certainty.▫ A coin is flipped twice.▫We conduct a 10 year medical study on

12,000 participants

Event

•The simplest outcome of an experiment•Toss two coins:

▫4 possible outcomes for each toss▫Each possible outcome is an event▫All the possible outcomes comprise the…

Sample space

•All the possible outcomes of an experiment: ▫(HH)▫(HT)▫(TH)▫(TT)

•More examples▫One coin▫One die▫Red, white, and blue socks

With and without replacement

Probability measure

•A function, P, from the subsets of the sample space (Ω) that satisfies the following axioms:1. P(Ω) = 1; that is, the sum of all the

probabilities in the sample space will = 12. P(A)≥0 for any event, A, that is included

in the sample space (Ω).3. For mutually exclusive events A1 and A2

within the same sample spaceP(A1UA2) = P(A1) + P(A2)

Classical Probability

•Proportion of times an event can be expected to occur

•All outcomes are equally likely:

Examples

•Flip a coin twice:▫P(HH) = P(HT) = P(TH) = P(TT) = 0.25

•Roll a die:▫P(1) = P(2) = P(3) = P(4) = P(5) = P(6) =

1/6

Examples•Consider events A and B (coins)•We can determine their probabilities by

adding the probability of each occurrence.▫A: exactly one head [(HT), (TH)]

P(A) = 0.25 + 0.25 = 0.50▫B: first flip head [(HH),(HT)]

P(B) = ¼ + ¼ = ½ •If we have an experiment where we roll

one die, what is the probability of rolling less than a 3?▫Sample space▫Event▫P(A)

Compound Events

•Mutually Exclusive Events▫If one event occurs, the other cannot

•Exhaustive Events▫Set of events that include all the possible

outcomes of an experiment▫Events are exhaustive because one of them

must occur▫When events are mutually exclusive and

exhaustive, sum of their probabilities must equal 1

Intersection and Union

•Intersection of Events▫Two or more events occur at the same time

in the same experiment. [A and B, or A and B and C]

•Union of Events▫At least one of a number of possible events

occurs in the same experiment. [A or B, or A or B or C]

What’s the probability?

Experiment: the throw of one dieA: (observe an even number)B: (observe a number <=3)

1. Describe A union B2. Describe A intersection B3. Calculate the probabilities of 1. and 2.

Example: Unions and Intersections

Age

Under 15(B)

15 or older(B’)

Male (A) 3477 5436 8913

Female (A’) 1249 1287 2536

4726 6723 11449

Addition Rules

•When events are mutually exclusive

•When events are not mutually exclusive

Practice

Access to

Internet?

Region

Northeast Midwest South West

Yes 9.7 11.8 16.9 12.2 50.6

No 1.2 2.3 3.8 2.1 9.4

10.9 14.1 20.7 14.3 60.0

Identity 2 events that are mutually exclusiveIdentify 2 events that intersect

Practice

What is the probability that a household would be in the South or Midwest or have internet access?What is the probability a household would be in the West and not have Internet access?

Access to

Internet?

Region

Northeast Midwest South West

Yes 9.7 11.8 16.9 12.2 50.6

No 1.2 2.3 3.8 2.1 9.4

10.9 14.1 20.7 14.3 60.0

A = 0.45

0.05

B = 0.15

What is the probability of Ac?What is the probability of Bc?What is the probability of AUB?

•Marginal Probability▫Probability a given event will occur. No

other events are considered. P(A)•Joint Probability

▫Probability that two or more events will all occur. P(A and B)

•Conditional Probability▫Probability that an event will occur given

that another event has already occurred. P(A|B)

Multiplication Rules

•Independent Events▫The occurrence of one has no effect on the

probability that the other will occur.•Dependent Events

▫The occurrence of one event influences the probability of the other.

Multiplication Rules

•When events are independent

•When events are not independent

Fireworks Chart

Age

Under 15(B)

15 or older(B’)

Male (A) 3477.304

5436.475

8913.779

Female (A’) 1249.109

1287.112

2536.221

4726.413

6723.587

114491.000

Conditional Probability

Type of Policy (%)

Category Fire Auto Other Total %

Fraudulent 6 1 3 10

Nonfraudulent 14 29 47 90

Total 20 30 50 100

Conditional Probability Problem

•A corporation is going to select 2 of its regional managers for promotion to VP. They have 6 male and 4 female regional managers. Assume each manager has an equal probability (1/10) of being selected.▫What is the probability that both people

selected for regional manager are male?

Practice Problems

•A fair coin is tossed 4 times. What is the probability of getting at least one tail?

•What is the probability of getting exactly one head?

•A card is drawn for a standard deck. What is the probability that card will be a jack or a king?

More Practice Problems

•A standard pair of 6-sided dice is rolled. What is the probability of rolling a sum greater than or equal to 3?

•Three cards are drawn with replacement from a standard deck. What is the probability that the 1st card will be a diamond, the 2nd card will be black, and the third card will be a queen?

Still more practice problems

•2 cards are drawn without replacement from a standard deck. What is the probability of choosing a club and then a black card?

•A box contains 6 green marbles and 19 white marbles. What is the probability of choosing a white marble if the first marble chosen was white?

Counting

•Principle of multiplication▫m ways for event 1 to happen▫n ways for event 2 to happen▫Total number of possibilities = m x n

•If each of k independent events can happen n different ways, the total number of possibilities is nk

Counting

•Factorial rule of counting▫n! = n x (n-1) x (n-2) x … x 1▫0! = 1

Counting

•Permutations: Number of possible arrangements of n items in order

Counting

•Combinations: Order doesn’t matter. We consider only the possible set of objects.

Counting Problems

•A 29-sided die is rolled 2 times. How many different outcomes are possible?

•License plates consist of 2 letters followed by three numbers. Duplicate digits are allowed. How many different outcomes are possible?

•A doctor visits her patients during morning rounds. In how many ways can she visit the 8 patients?

More Counting Problems

•A coordinator will select 6 songs from a list of 8 songs to compose a musical lineup. How many different lineups are possible?

•4 cards are chosen from a standard deck. How many different 4 card hands are possible?

•A person tosses a coin 16 times. In how many ways can he get 6 heads?

Practice

•The daily number in a state lottery is a 3-digit integer between 000 and 999. ▫What is the probability that the winning

number will be 555?▫Is the probability you found in part (a) an

example of classical, relative frequency, or subjective probability?

▫Today’s winning number is 347. You are going to buy a ticket tomorrow and you plan to select number 347. Is this a good idea? Why or why not?

PracticeYour company has two computer systems available for processing telephone orders. Computer system A has a 10% chance of being down; computer system B has a 5% chance of being down. The computer systems operate independently. At least one system needs to work in order to process new orders. For a typical telephone order, determine the probability that:

▫Neither computer system will be operational.

▫Both computer systems will be operational.▫Exactly one of the computer systems will

be operational.▫What is the probability that the order can

be processed without delay?

Practice

•A security service employing 10 officers has been asked to provide 3 officers for crowd control at a local carnival. In how many different ways can the firm staff this event?