Welcome to MM150 Unit 7

Seminar

DefinitionsExperiment - A controlled operation that yields a set of results.

Outcomes - The possible results from an experiment.

Event - A subcollection of the outcomes of an experiment.

Definitions ContinuedEmpirical Probability - The

relative frequency of occurrence of an event and is calculated by ACTUAL OBSERVANCE of an experiment.

Theoretical Probability - The determination of probability through a STUDY OF POSSIBLE OUTCOMES that can occur for the experiment.

Definitions ContinuedEmpirical Probability

P(E) = number of times event E has occurred total number of time exp. is performed

Theoretical Probability (Outcome equally likely)P(E) = number of outcomes favorable to E total number of possible outcomes

ExampleLet’s use 10 flips of a fair coin, which has 2 outcomes: heads or tails.

Empirical: I toss the coin 10 times and flip heads 7 times.P(Head) = 7/10 = 0.7

Theoretical: There are 2 outcomes, 1 of which is favorable to being a head.P(Head) = 1/2 = 0.5

Law of Large Numbers

Probability statements apply in practice to a large number of trials, not to a single trial. It is the relative frequency over the long run that is accurately predictable, not individual events or precise totals.

4 Important Facts1. The probability of an event that cannot occur is 0. It is said to be impossible.

2. The probability of an event that must occur is 1. It is said to be certain.

3. Every probability is a number between 0 and 1 inclusive; that is, 0 ≤ P(E) ≤ 1.

4. The sum of the probabilities of all possible outcomes of an experiment is 1. That is,P(A) + P(not A) = 1P(not A) = 1 - P(A)

Single Toss of Fair Coin

Outcomes - There are 2, heads or tails.

P(Head) = 1/2 = 0.5

P(Tail) = 1/2 = 0.5

P(Head) + P(Tail)

= 1/2 + 1/2

= 1

P(Tail) = 1 - P(Head)

P(Tail) = 1 - 1/2

P(Tail) = 1/2

Single Toss of Fair Coin

Outcomes - There are 2, heads or tails.

P(Head or Tail) = 2/2 = 1

P(Yellow) = 0/2 = 0

OddsOdds against event

P(event fails to occur) = P(failure) P(event occurs) P(success)

Odds in favor of event

P(event occurs) = P(success)P(event fails to occur) P(failure)

Odds ExampleLook at page 293 of your text, #49.The odds against the person having A+ blood.

Odds against A+ = P(person not having A+) P(person having A+)

= 66/34 or 66:34

Now, what if the question was what are the odds for a person having A+ blood. It would be34/66 or 34:66.

Probability from Odds

Again, turn to page 293 of your text, #47.

The probability that the person has A+ blood.

P(A+ Blood) = # of outcomes favorable to A+ total # of outcomes

= 34/100 = 0.34

NOTE Odds for was 34:66; therefore probability of A+ is 34/(34+66) = 34/100

Expected ValueE = P1*A1 + P2A2 + ... + PnAn

where P1 is the probability the first event will occur and A1 is the net amount won or lost if the first event occurs

P2 is the probability the second event will occur and A2 is the net amount won or lost if the second event occurs ...

Expected Value Example

Page 301 #16 In July in Seattle, the grass grows 1/2 in. a day on a sunny day and 1/4 in. a day on a cloudy day. In Seattle in July, 75% of the days are sunny and 25% are cloudy.B) Determine the expected total grass growth in the month of July in Seattle.

E = 0.75*1/2 + 0.25*1/4 = 0.375 + 0.0625 = 0.4375 This is the expected growth per day.

There are 31 days in July

(0.4375)(31) = 13.5625 inches is the expected growth in July in Seattle

DefinitionsSample Space - All possible outcomes of an experiment.

Sample Point - Each individual outcome in the sample space.

Counting Principle - If a first experiment can be performed in M distinct ways and a second experiment can be performed in N distinct ways, then the two experiments in that specific order can be performed in M*N distinct ways.

Tree Diagram ExampleMathy’s Pizza has a choice of:3 crusts: hand-tossed, pan, thin2 sauces: traditional, meat3 toppings: pepperoni, sausage, green peppers

Construct a tree diagram and list the sample space.What is the probability that a student selects meat sauce?

h

p

t

Crust Saucer

r

r

m

m

m

Toppinge

e

e

e

e

es

s

s

s

s

s

g

g

g

g

g

g

Sample Spacehrehrshrghmehmshmgpreprsprgpmepmspmgtretrstrgtmetmstmg

With/Without Replacement

A bag with 5 red marbles, 2 blue marbles and 1 white marble.

With ReplacementYou pick one marble out of the bag, record it and put it back in the bag. It is available

to be picked again.

Counting Principle: 8 * 8 = 64There are 64 outcomes.

For example:R1R1R1R2R1R3R1R4R1R5R1B1R1B2R1WR2R1R2R2R2R3R2R4R2R5R2B1R2B2R2W

...

Without ReplacmentYou pick one marble out of the bag, record it and set it aside (not back in the bag). It is not

available to be picked again.

Counting Principle: 8 * 7 = 56There are 56 outcomes.

For example:R1R2R1R3R1R4R1R5R1B1R1B2R1WR2R1R2R3R2R4R2R5R2B1R2B2R2W

...

Note: doubles missing

Or ProblemsP(Heart or 6) = P(Heart) + P(6) ---

Now...think about this...There is a 6 of hearts that will be counted in the hearts and with the 6s, so it is counted twice. We only want it to be counted once. Therefore we have to subtract 1/52 from the equation above.

P(Heart or 6) = P(Heart) + P(6) - 1/52 = 13/52 + 4/52 - 1/52 = 16/52

NOTE: P(Heart ∩ 6) = 1/52, soP(Heart or 6) = P(Heart) + P(6) - P(Heart ∩ 6)

Or ProblemsP(A or B) = P(A) + P(B) - P(A ∩ B)

When A and B are mutually exclusive

P(A or B) = P(A) + P(B)