PBIA - AP Statistics Liberty High School - Home

Statistics – Ch 5.3 Notes Name: _____________________ Conditional Probability and Independence

What is Conditional Probability? Let’s refer to our example from 5.2 using the table below. Define the events A = is male B = has pierced ears What if we were asked:

1. If we know that a randomly selected student has pierced ears, what is the probability that the student is male?

2. If we know that a randomly selected student is male, what is the probability that the student has pierced ears?

Calculating Conditional Probabilities Practice Problem: We classified the residents of a large apartment complex based on the events

A = reads USA Today B = reads NY Times. The complete Venn diagram is reproduced here: What is the probability that a randomly selected resident who reads USA Today also reads the NY Times? (Use probability notation)

The probability that one event happens given that another event is already known to have happened is

called a ________________________________________.

Suppose we know that event A has happened. Then the probability that event B happens given that

event A has happened is denoted by ________________.

Read | as “______________”

To find the conditional probability P(A | B), use the formula: ________________________________ The conditional probability P(B | A) is given by: __________________________________________

Key

PAIB 0.1845

PBIA0.2111

Conditional Probability

PBIA

given

PCA1BPCAPCB

PCBAPCBrayPLA

pBIA PCPB.fr 9i9oI 0.125


Calculating Conditional Probabilities Consider the two-way table on page 321. Define events E = the grade comes from an EPS course L = the grad is lower than a B. Find P(L) Find P(E | L) Find P(L | E) The General Multiplication Rule Practice Problem: The Pew Internet and American Life Project find that 93% of teenagers (ages 12 to 17) use the Internet, and that 55% of online teens have posted a profile on a social-networking site. Find the probability that a randomly selected teen uses the internet and has posted a profile. Show your work. Tree Diagrams The general multiplication rule is especially useful when a chance process involves a ________________ of outcomes. In such cases, we can use a tree diagram to display the sample space.

The probability that events A and B both occur can be found using the general multiplication rule

P(A ∩ B) = _________________

where P(B | A) is the conditional probability that event B occurs given that event A has already occurred.

Keg

tota630

70,60 6 0.3656 160210

totals 33922952 365610,008003656

0.2188 sooPLIE PGEf 1

ProbabilityHatHegradecomesfromanEpsgiventhatit is lowerthanaB PCE go.ie 18800Too 0.5

probabilitythatthegradeis below a B giventhatitcomesfromePs

PLA PBIA

D Online P o 0.93 P oANDP P o PPloD profile PPlo 0.55 0.93 0.55

0.5115

Theprobabilitythata randomlyselectedteen uses theinternetandhas posted a profileon a social networking site

is 0.5115

sequence


Example Problem Tree Diagram: The Pew Internet and American Life Project finds that 93% of teenagers (ages 12 to 17) use the Internet, and that 55% of online teens have posted a profile on a social-networking site.

x What percent of teens are online and have posted a profile? Practice Problem: On the 2011 Association of Tennis Professionals tour, Roger Federer made 63% of his first-serves. When he made his first-serve, he won 78% of the points. However, when he missed his first-serves and had to serve again, he won only 57% of the points.

1. Express each probability in probability notation.

2. Create a tree diagram to display all possible outcomes.

3. What is the probability that Federer makes the 1st serve and wins the point?

4. What is the probability that Federer wins the point?

Keg

online Profile Sampkspace_o.s PCOANDP

yesyes

Randomly no 0.4185selectteen

yo yes oooono

1 no 0.070

pmake 0.63 PCwinlmake 0.78 Pwinlmiss 0.57

1stServe Point Samplespacyo4914

Makewin

0.63 Lose 0.1386

misswin 0.2109

Lose 0.1591

pmakeAwin 0.63f781 0.4914

pflmakerwin U missnwin

0.631178 0.376.57

04914 f 2109

10770231


Conditional Probability and Independence When knowledge that one event has happened does not change the likelihood that another event will

happen, we say that the two events are ______________________________.

Proving Independence:

x If we are asked to prove that two events are independent, we must show that ______________

x Therefore, whenever P(B| A) ≠ P(B) we can conclude that events A and B are ________________

Example Problem:

Using our statistics example with the events A = a male student and B = a student with pierced ears.

1. Are the events “male” and “pierced ears” independent? Justify your answer Independent Practice: Is there a relationship between gender and handedness? To find out, we used CensusAtSchool’s Random Data Selector to choose an SRS of 100 Australian high school students who completed a survey. The two-way table displays data on the gender and dominant hand of each student. Are the events “male” and “left-handed” independent? Justify your answer

Two events A and B are independent if the occurrence of one event does not change the probability that the other event will happen. In other words, events A and B are independent if

P(A | B) = P(A) and P(B | A) = P(B).

Key

independent

PCBA PCB

notindependent

PBA 0.21 since pBIA PCB theseevents are notP B 101 0.5282 independentKnowingstudent ismaledecreases178 theprobabilitythatthestudenthaspiercedPLBIA t PB ears

0.211110.5787

M LH

PCLAIM PGH

6 Nothese events are notindependentKnowingthestudentismaleincreases theprobability0.1522 0.10 of thestudentbeing left handed


Independence: A Special Multiplication Rule

This rule only applies to events that are _________________________________! Practice Problem: Following the Space Shuttle Challenger disaster, it was determined that the failure of O-ring joints in the shuttle’s booster rockets was to blame. Under cold conditions, it was estimated that the probability that an individual O-ring joint would function properly was 0.977. Assuming O-ring joints succeed or fail independently, what is the probability all six would function properly?

Practice Problem: You are playing a game that requires rolling a dice 3 times consecutively. What is the probability that you roll a 3 on the first roll, a 6 on the second roll, and a 2 on the last roll?

Probability of “at least one” Practice Problem: Many people who come to clinics to be tested for HIV, the virus that causes AIDS, don’t come back to learn the test results. Clinics now use “rapid HIV tests” that give a result while the client waits. IN a clinic in Malawi, for example, use of rapid tests increased the percent of clients who learned their test results from 69% to 99.7% The trade-off for fast results is that rapid tests are less accurate than slower laboratory tests. Applied to people who have no HIV antibodies, one rapid test has probability about 0.004 of producing a false positive (that is, of falsely indicating that antibodies are present.) Problem: If a clinic tests 200 randomly selected people who are free of HIV antibodies, what is the chance that at least one false positive will occur?

Multiplication rule for independent events

If A and B are independent events, then the probability that A and B both occur is

P(A ∩ B) = P(A) • P(B)

Key

independent

s succeed p Is n ys n35 A4S A55A 6s0.977 0.922 0.977 0.922 0.922 0.977

0.97716

0.8697 istheprobabilitythatall six oringsfunctionproperly

P 3 6A2 p i p 6 p 2f te te1216 0.0046

0.996 negative0.004falsePositive

usecomplementRuledAtleasttoutcomespf.myl p a Patleast1 I P none 0.004 0.9960.996 0.996

p atleast1 I 0.996J 0.996560040.996 co.ma

0.996 0.9966.004 0.996

pCat 10 it0.996 0.9960.996 Co004ooo4 ooo40.996 0.996

o996oooo oof 0.996

i0.004 0.0040.004 0.004

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PBIA - AP Statistics Liberty High School - Home