5.1 Constructing Models of Random Behavior

Pgs 288 - 301

FUNDAMENTAL FACTS

Event – set of possible outcomes from a random situationProbability – number between 0 and 1 (0% - 100% that describes how likely an event is going to happen. 100% is high and 0% is low.

Notation: P(A) means Probability that event A happens

P(not A) = 1 – P(A) probability A does not happen.

FUNDAMENTAL FACTS

• An event A and not A are complements• Probability is determined as:

• Probability Distribution summarizes all possible outcomes and their probability (table, list, chart, etc

D1 on Pg 289

D1. Create a probability distribution for Jack’s values.

# Who Choose T Probability

0

1

2

Pails of water were soooo 2014

Jack and Jill create a taste test to see if people can determine the difference between bottled (B) water and tap (T) water.What’s being tested?

If no one can tell the difference, the results should be random chance and P(T) = 0.5

What if 2 tests are used simultaneously?Find P(TT):

J & J doing Math

Jack says: Three possible outcomes. No Tap One Tap All Tap

Jill says: Four OutcomesBB TB BT TT

D2 on Pg 289

• What is the sum of probability in a probability distribution? Why must it be so?

The sum must be 1 to include all outcomes. Also one of the outcomes has to occur.

D3 on pg 289

• Who do you think is right, Jack or Jill? Give your reason(s). How could Jack and Jill decide which of them is right?

Jill is correct. Jack forgot to account for the two ways that one T and one B can occur: TB & BT. Flip a distinct coin for each person which heads being T and tails being B many times!

Probabilities

• Observed Data (long-run relative frequencies) - thousands or more observations

• Symmetry (equally likely outcomes) – physics said they are as likely to occur

• Subjective estimates – no data to gather or list equally likely outcomes (F = 60% of possible scores after all) but I still have an answer to your question.

Sample Spaces

• Sample Space – complete list of disjoint outcomes and must total a probability of 1Disjoint – two different outcomes can’t occur on the same opportunity (Not quantum mechanics)Mutually Exclusive – two different outcomes can’t occur on the same opportunity (Not quantum mechanics)

Did you just make me copy the same definition twice??!!

D4 on pg 291

• Why is Jack really wrong? Using coins lets look:

• First flip is tails• First flip is heads, Second is tails• First flip is heads, Second is headsIs this list contain all outcomes? Are they disjoint? Are they equally likely? Complete and disjoint. Not Equal.

Law of Large Numbers

• Guarantees that relative frequencies will converge to the probabilites specified by the model as the sample size increases if the trial remain independent.– The penny will eventually start to approach

P(H) = 0.5

Misinterpretations of LoLN – being DUE

• Baseball batter is hitting .300 (roughly gets a hit 30% of the time. People often remark if the player has not had a hit in say last five attempts that he is “due” to get a hit this time

• Roulette is filled with constant “dueness” (can someone run to SEV and bring me a Diet Dew for 4th?) Odds of Red or Black are slightly less then 50%. People will bet that it’s due. If the wheel came out R-R-R-R-R-R-R … the probability of black is the same on the 7th spin as it is the first.

LoLN

• The point is that the LoLN will eventually even out. It is a HUGE gambler’s fallacy to hope they can outlast the casino for the next shift. They (not you) make hundreds of millions of dollars because they have the money to wait out these stat anomalies.

P(R-R-R-R-R-R-B) = P(R-R-R-R-R-R-R)

Fundamental Principal of Counting

• Multiply the possible outcomes of each stage together to determine the total number of possible outcomes.

• Tools: Two Way Tables – two stages (like Punnet squares in Genetics)

Tree diagram – multi stage!

D7 on pg 296

• J & J use three colas and have five people taste test. Options for each person are 1st, 2nd 3rd, No preference (copping out)

How many possible outcomes? Are they all equally likely?4 x 4 x 4 x 4 x 4 = 1024One cola probably sucks and who will say “No preference”?

D8 on 296

• Suppose you flip a fair coin seven times (AP answers alert! The use of fair is great diction for you to apply to your answers)

a) How many possible outcomes are there? b) What is the probability that you will get

seven heads?c) What is the probability you will get heads six

times and tails once?

D8 on 296 Answers

• A) 2^7 = 128 possible outcomes• B) 1/128• C) A tail could occur on anyone single one flip

for 7/128

Practice problems!

• Pg 296 – 298 #1-9

• Homework Pg 298-299 #1-3, 5, 7, 9

1st 2nd 3rd 4th Tap Selections

T T T T 4

T T T B 3

T T B T 3

T B T T 3

B T T T 3

T T B B 2

T B B T 2

T B T B 2

B T T B 2

B T B T 2

B B T T 2

T B B B 1

B T B B 1

B B T B 1

B B B T 1

B B B B 0

# Who Chose T Probability

0 1/16

1 4/16 = ¼

2 6/16 = 3/8

3 4/16 = ¼

4 1/16

• P1

• B) 1/16• C) Probably not. If no one can tell the

difference, still a 1/16 chance they get it all right. Larger sample size recommended.

P2A) 27 days listed on NWS forecast a low temp of

30, 13 days actually had a low temp of 30. Based on the estimate, 13/27 ~ 0.48

B)

C) There appears to be a bias toward predicting weather too warm. 10x colder than forecast

Actual Low Estimated Probability

20 2/27

25 8/27

30 13/27

35 3/27

40 1/27

P3 a) One solution:{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

B) Yes

C) 1/12

P4a) 28 & 35; 28 & 41, 28 & 47; 28 & 55; 35 & 41;

35 & 47; 35 & 55; 41 & 47; 41 & 55; 47 & 55b) Yesc) 1/10d) {28 & 55; 35 & 47; 35 & 55; 41 & 47; 41 & 55;

47 & 55} = 6/10

P5a) Yes, list is completeb) Not disjoint – Heads on first flip and Heads

on second flip can happen on same pairc) No, the event heads on second flip has the

same probability (1/2) as heads on first flip, but both are more likely than heads on neither flip (1/4)

P6a) Yes, you can list a sample space and it would

look like the 16 outcomes in P1, with T representing being right-handed and B representing being left-handed

b) No being left-handed and being right-handed are not equally likely, so you cannot determine the probability without more info about the percentage of students in the school who are right-handed

P7a) Since the proportion that were heads after

the first two spins was zero, the first and second spins must have been tails. The third was heads because the proportion that were heads went up. The fourth was tails and the fiftieth was tails

b) About 0.44

P8a)

b) Six

c) It is impossible to tell the probabilities with knowing the quality of the ice cream and the price. Still unlikely the outcomes are equal

P9a) There are 3 x 7 = 21 different pairs of dentists

and hygienistsb) The probability of getting your favorites pair

is 1/21

A B C

a aA aB aC

b bA bB bC

c cA cB cC

d dA dB dC

e eA eB eC

f fA fB fC

g gA gB gC