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The Practice of Statistics, 5th Edition
Starnes, Tabor, Yates, Moore
Bedford Freeman Worth Publishers
CHAPTER 5Probability: What Are the Chances?
5.3
Conditional Probability and Independence
The Practice of Statistics, 5th Edition 2
5.3 Reading Quiz: Matching
1. formula for calculating conditional probability
2. general multiplication rule
3. formula for determining independent events
4. multiplication rule for independent events
5. Can disjoint events be independent?
a) 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 ∗ 𝑃(𝐵)
b) 𝑃 𝐴 𝐵 =𝑃(𝐴∩𝐵)
𝑃(𝐵)
c) 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 ∗ 𝑃 𝐴 𝐵
d) 𝑃 𝐴 𝐵 = 𝑃(𝐴)
e) 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 ∗ 𝑃(𝐵|𝐴)
f) yes
g) no
The Practice of Statistics, 5th Edition 7
Try this
A recent census at a major university revealed that 40% of its students
mainly used Macintosh computers (Macs). The rest mainly used PCs. At the
time of the census, 67% of the school’s students were undergraduates. The
rest were graduate students. In the census, 23% of the respondents were
graduate students who said that they used PCs as their primary computers.
Suppose we select a student at random from among those who were part of
the census and learn that the student mainly uses a PC. Find the probability
that this person is a graduate student. Show your work.
𝑃 𝐺 𝑃𝐶 =𝑃(𝐺∩𝑃𝐶)
𝑃(𝑃𝐶)=
0.23
0.60= 0.383. Given that a student mainly uses a PC,
there is a 0.383 probability that the person is a graduate student.
0.10 0.370.23
0.30
Mac PC Total
Graduate Student
0.10 0.23 0.33
Undergraduate Student
0.30 0.37 0.67
Total 0.40 0.60 1.0
Learning Objectives
After this section, you should be able to:
The Practice of Statistics, 5th Edition 8
✓ USE the general multiplication rule to CALCULATE probabilities.
✓ USE tree diagrams to MODEL a chance process and CALCULATE
probabilities involving two or more events.
Conditional Probability and Independence
The Practice of Statistics, 5th Edition 9
The General Multiplication Rule
The probability that events A and B both occur can be found
using the general multiplication rule
P(A ∩ B) = P(A) • P(B | A)
where P(B | A) is the conditional probability that event B occurs
given that event A has already occurred.
General Multiplication Rule
In words, this rule says that for both of two events to occur, first one
must occur, and then given that the first event has occurred, the
second must occur.
The Practice of Statistics, 5th Edition 10
Playing in the NCAA
About 55% of high school students participate in a school athletic team
at some level, and about 5% of these athletes go on to play on a
college team in the NCAA
(http://www.washingtonpost.com/wp-
dyn/content/article/2009/09/23/AR2009092301947.html,
http://www.collegesportsscholarships.com/percentage-high-school-
athletes-ncaa-college.htm).
Problem: What percent of high school students play a sport in high
school and go on to play a sport in the NCAA?
Solution: We know P(high school sport) = 0.55 and P(NCAA sport |
high school sport) = 0.05, so P(high school sport and NCAA sport) =
P(high school sport) x P(NCAA sport | high school sport) = (0.55)(0.05)
= 0.0275. Almost 3% of high school students will play a sport in high
school and in the NCAA.
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Tree Diagrams
The general multiplication rule is especially useful when a chance
process involves a sequence of outcomes. In such cases, we can use a
tree diagram to display the sample space.
Consider flipping a coin
twice.
What is the probability of
getting two heads?
Sample Space:
HH HT TH TT
So, P(two heads) = P(HH) = 1/4
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Example: Tree Diagrams
P(online) = 0.93
P(profile | online) = 0.55
P(online and have profile) = P(online) × P(profile |online)
= (0.93)(0.55)
= 0.5115
51.15% of teens are online and have posted
a profile.
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.
What percent of teens are online and have posted a profile?
The Practice of Statistics, 5th Edition 13
Late for school
Shannon hits the snooze bar on her alarm clock on 60% of school days. If she
doesn’t hit the snooze bar, there is a 0.90 probability that she makes it to class
on time. However, if she hits the snooze bar, there is only a 0.70 probability
that she makes it to class on time. On a randomly chosen day, what is the
probability that Shannon is late for class?
The probability that she hits the snooze bar and is late is (0.60)(0.30) = 0.18. The
probability that she doesn’t hit the snooze bar and is late is (0.40)(0.10) = 0.04.
Because these outcomes are mutually exclusive, P(late) = 0.18 + 0.04 = 0.22.
Shannon is late on about 22% of school days.
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Late for school
Suppose that Shannon is late for school. What is the probability that
she hit the snooze bar that morning?
𝑃(ℎ𝑖𝑡𝑠 𝑠𝑛𝑜𝑜𝑧𝑒 𝑏𝑎𝑟 | 𝑙𝑎𝑡𝑒 𝑓𝑜𝑟 𝑠𝑐ℎ𝑜𝑜𝑙) =𝑃(ℎ𝑖𝑡𝑠 𝑠𝑛𝑜𝑜𝑧𝑒 𝑏𝑎𝑟 𝑎𝑛𝑑 𝑙𝑎𝑡𝑒 𝑓𝑜𝑟 𝑠𝑐ℎ𝑜𝑜𝑙)
𝑃(𝑙𝑎𝑡𝑒 𝑓𝑜𝑟 𝑠𝑐ℎ𝑜𝑜𝑙)
𝑃(ℎ𝑖𝑡𝑠 𝑠𝑛𝑜𝑜𝑧𝑒 𝑏𝑎𝑟 | 𝑙𝑎𝑡𝑒 𝑓𝑜𝑟 𝑠𝑐ℎ𝑜𝑜𝑙) =0.18
0.18 + 0.04= 0.818
When Shannon is late, there is a 0.818 probability that she hit the snooze bar that
morning.
Section Summary
In this section, we learned how to…
The Practice of Statistics, 5th Edition 15
✓ USE the general multiplication rule to CALCULATE probabilities.
✓ USE tree diagrams to MODEL a chance process and CALCULATE
probabilities involving two or more events.
Conditional Probabilities and Independence
The Practice of Statistics, 5th Edition 16
PAGE 334
74 - 80 EVEN
Homework