chapter 15imakemathmoreradicalvaldez.weebly.com/.../0/7/1/30713323/pp_not… · chapter 15...

appstats15.notebook January 26, 2015

Chapter 15

Probability Rules!

Objective:

1. Students will calculate conditional probabilities.2. Students will construct venn diagrams and tree diagrams.

Recall That…• For any random phenomenon, each trial generates an outcome.• An event is any set or collection of outcomes.• The collection of all possible outcomes is called the sample space, denoted S.

Events• When outcomes are equally likely, probabilities for events are easy to find just by counting.• When the k possible outcomes are equally likely, each has a probability of 1/k.• For any event A that is made up of equally likely outcomes,

pg 345


The General Addition Rule

• When two events A and B are disjoint, we can use the addition rule for disjoint events from Chapter 14:

P(A or B) = P(A) + P(B)• However, when our events are not disjoint, this earlier addition rule will double count the probability of both A and B occurring. Thus, we need the General Addition Rule.

pg 346General Addition Rule:• For any two events A and B,

P(A or B) = P(A) + P(B) – P(A and B) • The following Venn diagram shows a situation in which we would use the general addition rule:

Ex#1a. One card is drawn. What is the probability it is red or an ace? P(ace or red)=P(red) + P(ace) P(ace and red)

go over just checking and step by step pg 347348

The General Addition Rule:

P﴾A∪ B﴿ =

Special Case: if A and B are mutually exclusive (disjoint), then P﴾A∪ B﴿ =

The General Multiplication Rule

• When two events A and B are independent, we can use the multiplication rule for independent events from Chapter 14: •

P(A and B) = P(A) x P(B)• However, when our events are not independent, this earlier multiplication rule does not work. Thus, we need the General Multiplication Rule.

pg 350


The General Multiplication Rule (cont.)

• We encountered the general multiplication rule in the form of conditional probability.

• Rearranging the equation in the definition for conditional probability, we get the General Multiplication Rule:

For any two events A and B, P(A and B) = P(A) x P(B|A)

orP(A and B) = P(B) x P(A|B)

Drawing Without Replacement

• Sampling without replacement means that once one individual is drawn it doesn’t go back into the pool. • We often sample without replacement, which doesn’t matter too much when we are dealing with a large population. • However, when drawing from a small population, we need to take note and adjust probabilities accordingly.• Drawing without replacement is just another instance of working with conditional probabilities.

pg 354

b. Two cards are drawn without replacement. What is the probability that they are both aces?

c. What is the probability of getting 5 hearts in a row?

P(both cards are aces)= P(ace)P(ace/first draw was ace) = 4 3 = 12 52 51 2652P(5 Hearts)= 13 12 11 10 9 = .0005 52 51 50 49 48

Conditional Probability:From the General Multiplication Rule, we can derive the formula for conditional probability (note thatP(A) cannot equal 0 since we know that A has occurred):

When we want the probability of an event from a conditional distribution, we write P(B|A) and pronounce it “the probability of B given A.”

P﴾ A| B﴿ =

P﴾B | A﴿ =

The probability that event A occurs if we know for certain that event B will occur is called____________________ probability.

The ____________________ probability of A given B is denoted: _______________

conditional

conditional P(A/B)


It Depends… (cont.)

• To find the probability of the event B given the event A, we restrict our attention to the outcomes in A. We then find the fraction of those outcomes B that also occurred.

• Note: P(A) cannot equal 0, since we know that A has occurred.

pg 349 d. What is the probability it is red given that it is a heart? The card is selected at random

i. P(heart/red)= P(heart and red) = P(red)

ii. P(red/heart)= P(red and heart)= P(heart)

The First Three Rules for Working with Probability Rules

Make a picture.

Make a picture.

Make a picture.

Picturing Probabilities

• The most common kind of picture to make is called a Venn diagram.

• We will see Venn diagrams in practice shortly…


TEST for Independence

• RECALL:Independence of two events means that the outcome of one event does not influence the probability of the other.• With our new notation for conditional probabilities, we can now formalize this definition:• Special Case : A and B are independent whenever P(B|A) = P(B). (Equivalently, events A and B are independent whenever P(A|B) = P(A).)

pg 351


Example: When flipping a coin twice, what is the probability of getting heads on the second flipif the first flip was a head?Event A: getting head on first flipEvent B: getting head on second flipEvents A and B are independent since the outcome of the first flip does not change theprobability of the second flip, so…

P﴾ A| B﴿ =

If events A and B are ____________________ then knowing that event B will occur does not change theprobability of A so for independent events:

P﴾ A| B﴿ =P﴾A﴿

Recall: Independent ≠ Disjoint

Disjoint events cannot be independent! Well, why not?

Since we know that disjoint events have no outcomes in common, knowing that one occurred means the other didn’t. Thus, the probability of the second occurring changed based on our knowledge that the first occurred.

It follows, then, that the two events are not independent.A common error is to treat disjoint events as if they were independent, and apply the Multiplication Rule for independent events—don’t make that mistake.

pair share

Depending on Independence

• It’s much easier to think about independent events than to deal with conditional probabilities.

• It seems that most people’s natural intuition for probabilities breaks down when it comes to conditional probabilities.

• Don’t fall into this trap: whenever you see probabilities multiplied together, stop and ask whether you think they are really independent.

pg 352


jeans other Total

8 1112 5M

FT

a. What is the probability that a male wears jeans?

P(Jeans/M)= 12 17

b. What is the probability that someone wearing jeans is male?P( M/Jeans)= 12 20

d. Are Gender and Attire independent? P(Jeans/M)=P(Jeans) 12 = 20 17 36

Pair shareEx #2


Tree Diagrams

• A tree diagram helps us think through conditional probabilities by showing sequences of events as paths that look like branches of a tree.

• Making a tree diagram for situations with conditional probabilities is consistent with our “make a picture” mantra.

pg 355 Tree Diagrams (cont.)

• Figure 15.4 is a nice example of a tree diagram and shows how we multiply the probabilities of the branches together:

pg 357

Pair share


Ex #3Ovarian cancer, though dangerous, is very rare afflicting only 1 out 5000 women. The test is highly sensitive able to correctly detect ovarian cancer 99.97% of women who have the disease. However it is unlikely to be used as a screening test because the test gave false positive 5% of the time. Why is this such a big problem? What is the probability that a woman who test positive using a this method actually has ovarian cancer.

.0002

.9998

ovarian cancer

no ovariancancer

test positive

test negative

test positive

test negative

P( Ovarian Cancer/ Positive)=

.9997

.0003

Reversing the Conditioning

Reversing the conditioning of two events is rarely intuitive. Suppose we want to know P(A|B), but we know only P(A), P(B), and P(B|A).We also know P(A and B), since

P(A and B) = P(A) x P(B|A) From this information, we can find P(A|B):

pg 358


Reversing the Conditioning (cont.)

• When we reverse the probability from the conditional probability that you’re originally given, you are actually using Bayes’s Rule.

What Can Go Wrong?

• Don’t use a simple probability rule where a general rule is appropriate:• Don’t assume that two events are independent or disjoint without checking that they are.• Don’t find probabilities for samples drawn without replacement as if they had been drawn with replacement.• Don’t reverse conditioning naively.• Don’t confuse “disjoint” with “independent.”

pg 360

What have we learned?

• The probability rules from Chapter 14 only work in special cases—when events are disjoint or independent.• We now know the General Addition Rule and General Multiplication Rule.• We also know about conditional probabilities and that reversing the conditioning can give surprising results.

pg 361What have we learned? (cont.)

• Venn diagrams, tables, and tree diagrams help organize our thinking about probabilities.

• We now know more about independence—a sound understanding of independence will be important throughout the rest of this course.

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