Section 5.4 - Conditional Probability Objectives: 1.Understand the meaning of conditional...

Section 5.4 - Conditional Probability

Objectives:

1. Understand the meaning of conditional probability.

2. Learn the general Multiplication Rule:

P(A∩B) =P(A and B) =P(A)⋅P(B |A)

P(A∩B) =P(A and B) =P(B)⋅P(A |B)


Example: The Sinking of the TitanicDo the data support the phrase “Women and children first?”

• 711/2201, or 32% of all passengers survived.• 367/1731, or 21% of the males survived.• 344/470, or 73% of the females survived.• The chance of survival depends on the condition of whether the

passenger was male or female.

The notion that probability can change if you are given additional information is called conditional probability.

Display 5.39Gender

Male Female Total

Survived?

Yes 367 344 711

No 1364 126 1490

Total 1731 470 2201


Conditional Probability from the Sample Space

Example: The Titanic and Conditional Probability

Let S be the event that the passenger survived and let F be the event that the passenger was female. What is the probability of survival, given that the passenger was female?

Display 5.39Gender

Male Female Total

Survived?

Yes 367 344 711

No 1364 126 1490

Total 1731 470 2201

P(S | F) =P(survived| female) =344470

=0.732



Example: Sampling Without Replacement

When you sample without replacement from a small population, the probabilities for the second draw depends on the outcome of the first draw.

Suppose you randomly choose two cards from a standard deck of 52 cards. Suppose the first card chosen is a heart. What is the probability that the second card chosen will also be a a heart?



Example: Sampling Without Replacement

When you sample without replacement from a small population, the probabilities for the second draw depends on the outcome of the first draw.

Draw two cards from a standard deck of 52 cards. Suppose the first card chosen is a heart. What is the probability that the second card chosen will also be a heart? After the first heart is chosen, the sample space is changed. There are only 51 cards and 12 hearts remaining.

P(Heart) =1352

=14=0.250 P(Heart | Heart) =

1251

=0.235



D18. When you compare sampling with and without replacement, how does the size of the population affect the comparison? Conditional probability lets you answer the question quantitatively.

Imagine two populations of students, one large (N = 100) and one small (N = 4), with half of each population male. Draw random samples of size n = 2 from each population.

First, consider the small population.

Find P(2nd is M | 1st is M), assuming you sample without replacement. Then calculate the probability again, this time with replacement.



First, consider the small population (N = 4).


P(M ) =24=12=0.500

Without replacement : P(M |M ) =13=0.333

With replacement : P(M |M ) =24=12=0.500



Next, consider the large population (N = 100).




Next, consider the large population (N = 100).


P(M ) =50100

=12=0.500

Without replacement : P(M |M ) =4999

=0.495

With replacement : P(M |M ) =50100

=12=0.500



How would you describe the effect of population size on the difference between the two sampling methods?



How would you describe the effect of population size on the difference between the two sampling methods?

If the population size is large relative to the sample size, sampling without replacement is about the same as sampling with replacement.


The Multiplication Rule for P(A and B)

P(F and S) = 344 / 2201P(F and D) = 126 / 2201

P(M and S) = 367 / 2201P(M and D) = 1364 / 2201

Female

Male

P(M)= 1731 / 2201

P(F)= 470 / 2201

Female and Survived

Male and Survived

Female and Died

Male and Died

Survived

Died

Survived

Died

P(S|F) = 344 / 470P(D|F) = 126 / 470

P(S|M) = 367/ 1731P(D|M) = 1364 / 1731



P(A∩B) =P(A)⋅P(B |A)

P(A∩B) =P(B)⋅P(A |B)



Event Probability

A and B P(A) P(B|A)P(B|A) B

P(A) A

P(B) B

P(A|B) AB and A P(B) P(A|B)

1st Event 2nd Event



Example:

P(F∩S) =P(F )⋅P(S|F )

=4702201

⋅344470

=3442201

=0.156


The Definition of Conditional Probability

For any two events A and B such that P(B) > 0,

P(A |B) =P(A∩B)

P(B)


The Definition of Conditional Probability

Example: Rolling Dice

Find the probability that you get a sum of 8, given that you rolled doubles.

P(D) =636

P(8 ∩D) =136

P(8 |D) =P(8 ∩D)

P(D)=

136636

=16


Conditional Probability and Medical Tests

Screening tests give an indication of whether a person is likely to have a particular disease or condition. A two-way table is often used to show the four possible outcomes of a screening test:

Test Result

Positive Negative Total

Disease

Present a b a+b

Absent c d c+d

Total a+c b+d a+b+c+d



The effectiveness of screening tests is judged using conditional probability.

Positive predicted value (PPV ) =P(disease|test positive)

=a

a+ c

Negative predicted value (NPV) =P(no disease|test negative)

=d

b+d



The effectiveness of screening tests is judged using conditional probability.

Sensitivity=P(test positive|disease)

=a

a+b

Specificity=P(test negative|no disease)

=d

c+d



A false positive is a positive test result when the patient does not have the disease or condition.

A false negative is a negative test result when the patient has the disease or condition.

P( false positive) =P(test positive|no disease)

=c

c+d

P( false negative) =P(test negative|disease)

=b

a+b


E51. A screening test for the detection of a certain disease gives a positive result 6% of the time for people who do not have the disease. The test gives a negative result 0.5% of the time for people who do have the disease. Large scale studies have shown that the disease occurs in about 3% of the population.

Fill in a two-way table showing the results expected for every 100,000 people.

Test Result


DiseaseYes

No

Total 100,000


E51. A screening test for the detection of a certain disease gives a positive result 6% of the time for people who do not have the disease. The test gives a negative result 0.5% of the time for people who do have the disease. Large scale studies have show that the disease occurs in about 3% of the population.

Fill in a two-way table showing the results expected for every 100,000 people.

Test Result


DiseaseYes 2,500 500 3,000

No 6,000 91,000 97,000

Total 8,500 91,500 100,000


What is the probability that a person selected at random tests positive for this disease?

What is the probability that a person selected at random who tests positive for the disease does not have the disease?

P(test positive) =

P(no disease | test positive) =

Test Result


DiseaseYes 2,500 500 3,000

No 6,000 91,000 97,000

Total 8,500 91,500 100,000


What is the probability that a person selected at random tests positive for this disease?

What is the probability that a person selected at random who tests positive for the disease does not have the disease?

P(test positive) =8,500

100,000=0.085

P(no disease | test positive) =6,0008,500

=0.705

Test Result


DiseaseYes 2,500 500 3,000

No 6,000 91,000 97,000

Total 8,500 91,500 100,000


Conditional Probability and Statistical Inference

To calculate a probability, you must work from a model.

For example, what is the probability of observing an even number of dots on a roll of a die?

The model is that the die is fair.

Using this model, P(even|fair die) = 3/6 = 1/2 = 0.5000.

Suppose you have a die that you suspect isn’t fair. How can you discredit the model that the die is fair?

Suppose you roll the die 10 times and get an even number every time. P(10 evens) = (1/2)10 = 0.00098.

The outcome is so unlikely that the model of a fair die should be rejected.

Section 5.4 - Conditional Probability Objectives: 1.Understand the meaning of conditional...

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