activity: the 1 in 6 wins game grab a...

chapter 5 Probability.notebook

1

January 12, 2016

Activity: The "1 in 6 Wins" Game

Grab a die!

0 1 2 3 4 5 6

# of winners


2

January 12, 2016

Chapter 5: Probability

The probability of any outcome of a chance process is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions.

> chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run

> probability describes what happens in very many trials


3

January 12, 2016

Law of Large Numbers: If we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value, which we call the probability of that outcome.


4

January 12, 2016

Check Your Understanding


5

January 12, 2016

Friday December 11th


6

January 12, 2016


7

January 12, 2016

Simulation: The imitation of chance behavior, based on a model that accurately reflects the experiment under consideration.

Step 1: (state the problem or describe the experiment)• Toss a coin ten times. What is the likelihood of a run of 3 or

more consecutive heads or tails?

Step 2: (state the assumptions)• A head or tail is equally likely to occur on each toss • Tosses are independent of each other

Step 3: (assigning labels)• One digit will represent one toss of the coin.• Odd digits represent heads, even digits represent tails• Using a random number generator, we "toss a coin" 10 times.• randint(1,10): 1,3,5,7,9 indicate tossing a heads and 2,4,6,8,10

indicates tossing a tails.

Step 4: Simulate many repetitions (we will do 25)

Step 5: State your conclusions


8

January 12, 2016

Trial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Run of 3? N Y Y Y Y Y N Y Y Y N Y Y Y N Y Y Y Y N Y N N Y Y

Nascar Cards and Cereal BoxesIn an attempt to increase sales, a breakfast cereal company decides to offer a NASCAR promotion. Each box of cereal will contain a collectible card featuring one of these NASCAR drivers: Jeff Gordon, Dale Earnhardt Jr., Tony Stewart, Danica Patrick, or Jimmie Johnson. The company says that each of the 5 cards is equally likely to appear in any box of cereal. A NASCAR fan decides to keep buying boxes of the cereal until he has all 5 drivers' cards. He is surprised when it takes him 23 boxes to get the full set of cards. Should he be surprised? Design and carry out a simulation to help answer this question.

State: What is the probability that it will take 23 or more boxes to get a full set of 5 NASCAR collectible cards?

Plan: Since each of the cards is equally likely to get chosen, we will assign a number 15 to each driver. Let 1=Gordon, 2=Jr., 3=Stewart, 4=Patrick, 5=Johnson. We will use randint(1,5) to simulate a card from one box of cereal. We will keep pressing enter until we get a full set of cards, meaning 1, 2, 3, 4 and 5. We will record how many times we hit enter until this happens, that will simulate how many boxes had to be purchased.


9

January 12, 2016

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

# of Boxes

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

# of Boxes


10

January 12, 2016

Nascar Cards and Cereal BoxesWhat if the cereal company decided to make it harder to get some drivers' cards than others? For instance, suppose the chance that each card appears in a box of cereal is: Jeff Gordon, 10%; Dale Earnhardt Jr. 30%; Tony Stewart, 20%; Danica Patrick, 25%; and Jimmie Johnson, 15%. How would you modify the simulation in the example to estimate the chance that a fan would have to buy 23 or more boxes to get the full set?

Read pg. 295299

HW 5.2 #1519 odd, 2023, 25


11

January 12, 2016

Monday December 14th

The sample space S of a random phenomenon is the set of all possible outcomes.

> When we toss a coin, there are two possible outcomes, heads and tails. The sample space is

An event is any outcome or a set of outcomes from some chance process. An event is a subset of the sample space.

> Events are usually designated by capital letters.

• A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space S and a way of assigning probabilities to events


12

January 12, 2016

Let's say there are two fair, six-sided die, each a different color in a particular board game. • There are 36 possible outcomes, so the following figure

shows our sample space.• Each outcome has a probability of 1/36

• If A is any event occurring, we write it's probability as P(A)

Example 1: What is the probability that the sum of two, fair, six-sided die of different colors is 5?

Let A="sum is 5"

There are four outcomes:

Therefore, P(A)=4/36

Example 2: Using the same dice as example 1, what is the probability that the sum isn't 5?

Let B="sum isn't 5"

There are 32 outcomes where the sum isn't 5, so...

P(B)=P(sum isn't 5)=P(not A)= 32/36

* Notice that P(A)+P(B)=1.


13

January 12, 2016

Example 3: Let's look at one more event, C: sum is 6.

P(C)=5/36

Example 4: What's the probability that we get a sum of 5 OR 6?

Because these events have nothing in common, we can add the probabilities of the individual events:

P(sum is 5 or sum is 6)=P(sum is 5)+P(sum is 6)=4/36+5/36=9/36

So, P(A or C)=P(A)+P(C)

Determining # of Outcomes

Let's say we are tossing a coin and rolling a die at the same time. What is the sample space? (i.e. how many possible outcomes are there?)

Tree Diagram:

H

T

123456

123456

H1

T1

Multiplication Principle:

If you can do one task in a ways and a second task in b ways, then both tasks can be done in a x b ways.


14

January 12, 2016

Example 5: Suppose you flip a coin four times.

A) How many possible outcomes are there when you flip four coins?

B) If we are only interested in the number of heads when tossing four coins, what is the sample space?

Probability Rules

1. The probability of any event is a number from 0 to 1.>

2. All possible outcomes together have a probability of 1.>

3. The probability that an event does not occur is 1 minus the probability that the event does occur.> The complement of any event A is the event that A does not

occur, written as AC. > The complement rule states:

4. If two events have no outcomes in common (mutually exclusive, or disjoint), the probability that one OR the other occurs is the sum of their individual probabilities. > The addition rule for disjoint events:


15

January 12, 2016

Read pg. 305309

HW 5.3 #27, 31, 32, 3947 odd


16

January 12, 2016

Tuesday December 15th

The following table displays the probability that college students in a particular age group are taking a distance-learning course for credit.

Find the probability that the chosen student is not in the traditional college age group (18 to 23).

Use both the Complement Rule and the Addition Rule for Disjoint Events. (solve two ways)

Example:

Here is data on the number of pierced ears in a college statistics class. There is data on two variables: gender, and whether or not the student has their ears pierced.

Suppose we choose a student at random from the class. Find the probability that that student...

a) Has pierced ears

b) is a male with pierced ears

c) is a male OR has pierced ears


17

January 12, 2016

Set Notation How it is read What it means"A union B" Set of all

outcomes that are either in A OR B

"A or B"

"A intersect B" "A and B"

"empty event" The event has no outcomes in it.

Set of all outcomes that are

in A AND B.

Mutually Exclusive (Disjoint): Two events are mutually exclusive if they have no outcomes in common and so can never occur together.

General Addition Rule for two events:

If A and B are any two events resulting from some chance process, then P(A or B) = P(A) + P(B) P(A and B)


18

January 12, 2016

Venn Diagrams

A and its complement Mutually exclusive events, A and B

Union of events A and B Intersection of events A and B


19

January 12, 2016


20

January 12, 2016


Read pg. 309314

HW 5.4 #29, 3336, 4955odd


21

January 12, 2016

Monday January 4th

1. What does mutually exclusive mean?

2. What is the general addition rule?

a) Find P(is male given he have ears pierced)

b) Find P(has pierced ears given he's a male)

Example 1:


22

January 12, 2016

• "given that" signifies condition

ex:


23

January 12, 2016

Example 2: What's the probability that a randomly selected resident who reads USA Today also reads the NY Times?

Read pg. 287295

HW 5.1 # 15, 711 odd

Read pg. 295299

HW 5.2 #1519 odd, 2023, 25

Read pg. 305309

HW 5.3 #27, 31, 32, 3947 odd

Read pg. 309314

HW 5.4 #29, 3336, 4955odd


24

January 12, 2016

Wednesday January 6th


25

January 12, 2016

Example 3:

Example 4: Tennis great Roger Federer made 63% of his first serves in the 2011 season. When Federer made his first serve, he won 78% of the points. When Federed missed his first serve and had to serve again, he won only 57% of the points. Suppose we randomly choose a point on which Federer served.

Sketch a tree diagram that can be used to find probabilities surrounding this problem.


26

January 12, 2016

Example 5: What's the probability that Federer makes the first serve and wins the point?

Example 6: What's the probability that he wins the point?

Read pg. 318326

HW 5.5 #5760, 6379odd


27

January 12, 2016

Thursday January 7th

Example 7:


28

January 12, 2016

Example:


29

January 12, 2016

** Two mutually exclusive events can never be independent.

Example: In baseball, a perfect game is when a pitcher doesn't allow any hitters to reach base in all nine innings. Historically, pitchers throw a perfect inning an inning where no hitters reach base about 40% of the time. So, to throw a perfect game, a pitcher needs to have nine perfect innings in a row. What is the probability that a pitcher throws nine perfect innings in a row? Assuming the pitcher's performance in an inning is independent of his performance in other innings?


30

January 12, 2016


Read pg. 326333

HW 5.6 # 8185odd, 8995odd, 9799


31

January 12, 2016

Example 8:

Review:

Page 340 #R5.1R5.10 and T5.1T5.14


32

January 12, 2016

The powerball continues to not have a winner, so the winnings are up to 1.3 billion dollars. Five numbers are drawn from a tumbler with 59 white balls and the final "powerball" is drawn from a different tumblr with 35 red balls. The powerball is won by matching five white balls' numbers and one red balls' numbers. Each of the six white balls are not replaced. There are 59 white balls and 35 red balls.

What are the chances of winning the powerball and matching all six numbers?

Monday Jan. 11th

activity: the 1 in 6 wins game grab a...

Documents