Assignments:

a.  3, 6, 10, 12

b.  15, 19, 23, 26, 27

c.  28, 33, 36, 38

d.  41, 42, 43, 47

e.  50, 51, 53, 55

f.  64, 68, 73

g.  75 and Activity 6.3: The “Hot Hand” in Basketball

Objectives:

Know how to find probability as it relates to area.

Know the difference between classical and experimental probability.

Know the probability formulas from the formula sheet and what they are for.

Know how to give the practical interpretation of a stated probability. What will happen in the long run? This is also called the relative frequency interpretation.

Know how to do the probability of “at least one”.

Know how to assign digits to create a simulation of a probability experiment. This involves the concepts of replacement and non-replacement, independence and dependence.

Know the relationships between mutually exclusive (disjoint), not mutually exclusive (not disjoint), independent, and dependent. Are disjoint events with probabilities greater than zero independent or dependent?

Know how to compute simple “and” and “or” probabilities.

Define disjoint events.

Set up tables to calculate probabilities.

Set up trees to calculate probabilities.

Find probability using relative frequency.

What are the “magic words” when dealing with relative frequency?

Use tree diagrams and tables in conjunction to find probabilities, including conditional probabilities.


Basic Terms

Chance Experiment: ______

Sample Space: ______

Event: ______

Simple Event: ______

Union: ______

Intersection: ______

Complementary Events: ______

Disjoint (Mutually Exclusive) Events: ______

Venn Diagrams

  1. Draw a Tree Diagram for all the possible outcomes of first tossing a coin and then rolling a 6-sided die.
  1. Do assignment a: 3, 6, 10, 12

Approaches to probability

Classical

  1. What is the probability of tossing tails and then rolling a 5?
  1. If I randomly draw a ball from a box containing 3 orange balls and 5 green balls, what is the probability that I will draw an orange ball? What is the probability that I will not draw an orange ball?
  1. If I randomly throw a dart at a target with three scoring areas consisting of concentric circles with radii of 2cm, 4cm and 6cm, what is the probability of the dart landing in the outer ring?

Relative Frequency

5

The Law of Large Numbers: ______

  1. Toss a coin 100 times and graph the relative frequency of heads after each toss.

Subjective

A personal measure of strength of belief.

  1. What do you believe is the probability of Oklahoma beating Oklahoma State in football this year?
  1. What do you believe are your chances of making a 3 or better on the AP Statistics exam?

Basic properties of probability:

Remember all probabilities are between 0 and 1.

If S is the sample space (every outcome possible) then P(S) = 1.

Complementary events have probabilities that add up to 1. P(-E) = 1 – P(E)

Do assignment b: 15, 19, 23, 26, 27

Addition Rule (disjoint events)

If two events E and F are disjoint, then P(EÈF) = (E or F) = P(E) + P(F).

  1. What is the probability of rolling a 3 on a pair of fair dice? What is the probability of rolling a 7? What is the probability of rolling a three or a 7?
  1. If I draw a card from a well shuffled deck. What is the probability of drawing a 2 or a face card?
  1. Are drawing a king and drawing a red card from a deck of well shuffled cards disjoint events?
  1. If you draw a card and roll 2 6-sided dice, are drawing an ace and rolling an 8 disjoint events?

Addition to the Addition Rule

If two events E and F are not disjoint, then…

P(E or F) = P(E) + P(F) – P(E and F)

  1. What is the probability of rolling an even number or a number greater than 3 on a 6-sided die?
  1. What is the probability of drawing either a spade or a face card?

Conditional Probability

What is the probability that an extra on Star Trek dies given that he is wearing a red shirt? – This is an example of a conditional probability.

The probability of event E occurring given that event F has already occurred can be found either by direct reasoning or by the formula:

  1. What is the probability of drawing a King of Spades from a well shuffled deck?
  1. What is the probability of drawing a King given that you’ve drawn a spade from a well shuffled deck? (Or…If you draw a spade what is the probability that it is a king?)
  1. Given the following table of values from a survey of teachers on job satisfaction…

·  Find the probability associated with each cell.

·  Find the probability that a teacher is satisfied with his job given that he is a high school teacher.

·  Find the probability that a teacher is a high school teacher given that he is satisfied with his job.

Satisfied / Unsatisfied / Total
College / 74 / 43 / 117
High School / 224 / 171 / 395
Elementary / 126 / 140 / 266
Total

Do assignment c: 28, 33, 36, 38

Independence

Given events E and F, E and F are independent if P(E|F) = P(E). That is to say the probability of E occurring remains the same whether F has occurred or not.

Are the events independent?

E = Mr. Moore is out of the room during a televised OU game.

F = OU makes a good play.

Since P(Mr. Moore is out of the room during a televised OU game) does not change whether OU makes a good play or not, E and F are independent.

Note that if P(E|F) = P(E) then P(F|E) = P(F). Independence works both ways.

Also, if E and F are independent and F happens or doesn’t happen, either way, it doesn’t effect the probability of E happening or not happening for that matter. If E and F are independent nothing we know about F effects anything involving the likelihood of E and vise-versa. So…

P(E|-F) = P(E), P(-E|F) = P(-E), and P(F|-E) = P(F), P(-F|E) = P(-F)

Multiplication Rule (independent events)

Given independent events E and F, the P(EÇF) = P(E and F) = P(E)*P(F)

1.  The probability that Mr. Moore will be out of the room during a televised OU game is .16. The probability that OU will make a great play is .37. Find the probability that OU will make a great play and Mr. Moore will be out of the room during a televised OU game.

2.  If you roll a 6-sided die 5 times, what is the probability that you roll the same number every time?

3.  What is the probability of drawing an even numbered card from a well shuffled deck and rolling a 10 on a pair of dice?

4.  If I randomly pick a student from our class and have them roll a pair of dice, what is the probability that a guy will roll “snake-eyes!”?

5.  Are disjoint events independent?

6.  If I draw 5 cards in row without replacing the cards each time, are the draws independent? If I replace the cards and reshuffle each time are they independent?

7.  P(rolling a 4 on a die and drawing a face card and tossing heads) =

Do assignment d: 41, 42, 43, 47
General Probability Rules

Addition Rule

For events E and F,

P(E or F) = P(E) + P(F) – P(E and F)

Example: 60% of all students eat lunch in the main lunchroom. 40% eat lunch in the Commons. 25% of all students eat lunch both places. Find the probability that a randomly selected student eats lunch either in the lunchroom or the Commons.

P(lunchroom) = .6 P(Commons) = .4 P(lunchroom and Commons) = .25

So…

P(lunchroom or Commons) =

P(lunchroom) + P(Commons) – P(lunchroom and Commons) = .6 + .4 - .25 = .75

1.  What is the probability that a student doesn’t each lunch either place?

2.  What is the probability that a student eats in only one of the two places?

Multiplication Rule

For events E and F,

P(E and F) = P(E|F)*P(F)

Example: 30% of those attending prom last year did so without a date. Of those without dates, only 13% actually asked someone to go with them as a date. If a student attending prom last year were randomly selected, what is the probability that he or she didn’t have a date and asked someone to prom? (These are not independent. Why?)

P(dateless) = .3 P(asked|dateless) = .13 P(asked) = unknown

We can do either

P(dateless and asked) = P(dateless|asked)P(asked)

or

P(asked and dateless) = P(asked|dateless)P(dateless)

Since we don’t know all of the info for the first method we’ll use the second.

P(asked and dateless) = .13*.3 = .039

Thus 3.9% of the students at prom asked someone to prom and were dateless on prom night.


Using a Tree Diagram to calculate probabilities.

When you use a tree diagram you can multiply along the branches to calculate the end probabilities. For example…

P(heads and 6) =

P(tails and 4 or heads and 2) =

65% of the senior class at MHS has contracted senioritis. A new test has been devised to diagnose this debilitating disease. The test isn’t perfect, however, returning false positives 2% of the time and false negatives 1.5% of the time.

1.  Draw the tree diagram for this scenario.

2.  If I select a senior at random find…

a.  P(correct diagnosis)

b.  P(false diagnosis)

3.  Twenty percent of all passengers who fly from Los Angeles to New York do so on Airline G. This airline misplaces luggage for 10% of its passengers, and 90% of this lost luggage is subsequently recovered. If a passenger who has flown from LA to NY is randomly selected, what is the probability that the selected individual flew on Airline G, had luggage misplaced, and subsequently recovered the misplaced luggage?

Do assignment e: 50, 51, 53, 55


Conditional Probability, Tables & Tree Diagrams

1)  A survey of MHS students found that 36% said that they would be interested in going to Saturn. Of those who wanted to go to Saturn, 60% were not seniors. Of those who did not want to go to Saturn, 30% were seniors.

Create a tree diagram for this situation.

What is the probability that a randomly selected

(a)  Student wanted to go to Saturn?

(b)  Was a senior and wanted to go to Saturn?

(c)  Student was a senior?

(d)  Senior wanted to go to Saturn?

(e)  Saturn wannabe was a senior?

Fill in the two-way (contingency) percent table with the information.

If 500 students were surveyed, fill in the two-way (contingency) counts table with the information.

2)  When the male students at MHS were asked, 50% said they do not date someone from MHS. When the female students were asked, 40% said they do not date someone from MHS. The male students make up 52% of the student population.

Draw a tree diagram to represent this situation.

Fill in the two-way (contingency) percent table with the information.

What is the probability that a randomly selected

(a)  Student does not date someone from MHS?

(b)  Student is female?

(c)  Student is female and does not date someone from MHS?

(d)  Student who dates someone from MHS is male?

If 500 students were surveyed, fill in the two-way (contingency) counts table with the information.

(e)  How many were male?

(f)  How many were females who did not date someone from MHS?

3)  Mr. Moore’s Statistics classes collected information on their gender and handedness. The two-way table below gives the percents of the statistics students who fall into each category.

If a student is chosen at random from the Statistics classes, what is the probability of a

(a)  Student being left-handed, if the student is a male?

(b)  Student being female, if the student is right-handed?

(c)  Student being female and right-handed?

If there are 140 Statistics students, (I’m dreaming now) complete the two-way counts table below.

(d)  How many are male and left-handed?

Draw a tree diagram with correct probabilities to represent this situation? There are two possible tree diagrams depending on which event you chose for your initial branches. Challenge: Work both!

4)  The following counts were provided concerning people in rural, suburban, and urban areas and the length of their vacations (1-7 days or 8+ days).

Complete the two-way percent table with the information.

What is the probability that a randomly selected

(a)  Person spends 8+ days on vacation, given that he/she is from a rural area?

(b)  Person is from a rural area, given that he/she spends 8+ days on vacation?

(c)  Person spends 1-7 days on vacation, given that he/she is from a suburban area?

(d)  Person is from an urban area, given that he/she spends 1-7 days on vacation?

Can you draw a tree diagram with correct probabilities to represent this situation?

5)  We wish to look at probabilities concerning whether students own a dog or have a cell phone. We poll 400 students at MHS. One hundred thirty-two own a dog. Eighty-four students owning a dog have a cell phone. One hundred forty students without a dog have a cell phone.

Complete the two-way table of counts for this situation.

Complete the two-way percent table for this situation.