Conditional Probability (1) by using statistics.pptx

Conditional Probability
The probability we assign to an event can change if we know that some
other event has occurred. This idea is the key to many applications of
probability.

The probability of an event based on the fact that some other event has
occurred, will occur, or is occurring.
P(B|A) =
The probability of event B occurring given that event A has occurred is
usually stated as “the conditional probability of B, given A; P(B|A)
Conditional Probability – Events Involving “And”


)
(
)
(
A
P
B
A
P
)
(
)
(
A
P
B
and
A
P

Example:
Given a family with two children, find the probability that both are boys,
given that at least one is a boy.
P({gb, bg, bb}) =
P(A and B) =
A = at least one boy
P(A) =
3/4
P({gb, bg, bb}  {bb}) = P({bb}) =
= 1/3
1/4
Conditional Probability P(B|A) =
3/4
1/4
B = both are boys
S= {gg, gb, bg, bb}
A = {gb, bg, bb}
B = {bb}
)
(
)
(
A
P
B
and
A
P
)
(
)
(
A
P
B
and
A
P
=

Example:
A single card is randomly selected from a standard 52-card deck. Given
the defined events A and B,
A: the selected card is an ace, B: the selected card is red,
find the following probabilities.
a) P(B) =
Independent Events
b) P(A and B) =
a) P(B) b) P(A and B) c) P(B|A)
= 1/2
P({Ah, Ad, Ac, As}  {all red}) = P({Ah, Ad}) = 2/52
Events A and B are independent as P(B) = P(B|A).
c) P(B|A) =
P(A)
P(A and B)
4/52
2/52
1/2
= =
52
26

• Consider tossing a fair six-sided die once and define events A = {2, 4,
6}, B = {1, 2, 3}, and C = {1, 2, 3, 4}. Calculate the following:
• P(A)
• P(A|B)
• P(A|C)
• Are A and B dependent or independent? Explain
• Are A and C dependent or independent? Explain

If A and B are any two events then
P(A and B) = P(A)  P(B|A)
Multiplication Rule of Probability - Events Involving “And”
11.3 – Conditional Probability – Events Involving “And”
P(A and B) = P(A)  P(B)
If A and B are independent events then
A jar contains 4 red marbles, 3 blue marbles, and 2 yellow marbles.
What is the probability that a red marble is selected and then a blue one
without replacement?
P(Red and Blue) = P(Red)  P(Blue|Red)
= 4/9  3/8
= 12/72
= 1/8 = 0.1667
Example:

Multiplication Rule of Probability - Events Involving “And”
11.3 – Conditional Probability – Events Involving “And”
A jar contains 4 red marbles, 3 blue marbles, and 2 yellow marbles.
What is the probability that a red marble is selected and then a blue one
with replacement?
P(Red and Blue) = P(Red)  P(Blue)
= 4/9  3/9
= 12/81
= 4/27 = 0.148
Example:

Example:
A number from the sample space S = {2, 3, 4, 5, 6, 7, 8, 9} is randomly
selected. Given the defined events A and B,
A: selected number is odd, and
B: selected number is a multiple of 3
find the following probabilities.
a) B = {3, 6, 9}
b) P(A and B) =
a) P(B) b) P(A and B) c) P(B|A)
P(B) = 3/8
P({3, 5, 7, 9}  {3, 6, 9})
= P({3, 9}) = 2/8 = 1/4
c) Probability of B given A has occurred:
P(B|A) =
P(A)
P(A and B)
4/8
1/4
1/2
= =

The table shows the results of a class survey.
Find P(own a pet | female)
The condition female limits the sample space to 14 possible outcomes.
Of the 14 females, 8 own a pet.
Therefore, P(own a pet | female) equals . 8
14
yes no
female 8 6
male 5 7
Do you own a pet?
14 females; 13
males

The table shows the results of a class survey.
Find P(wash the dishes | male)
The condition male limits the sample space to 15 possible outcomes.
Of the 15 males, 7 did the dishes.
Therefore, P(wash the dishes | male) 7
15
yes no
female 7 6
male 7 8
Did you wash the dishes last night?
13 females; 15
males

Using the data in the table, find the probability that a sample of not
recycled waste was plastic. P(plastic | non-recycled)
The given condition limits the
sample space to non-recycled
waste.
Material Recycled Not Recycled
Paper 34.9 48.9
Metal 6.5 10.1
Glass 2.9 9.1
Plastic 1.1 20.4
Other 15.3 67.8
The probability that the non-recycled waste was plastic is about 13%.
=
20.4
156.3
0.13
A favorable outcome is
non-recycled plastic.
P(plastic | non-recycled) =
20.4
48.9 + 10.1 + 9.1 + 20.4 + 67.8
Let’s Try One

Researchers asked people who exercise regularly whether they jog or
walk. Fifty-eight percent of the respondents were male. Twenty
percent of all respondents were males who said they jog. Find the
probability that a male respondent jogs.
Relate: P( male ) = 58%
P( male and jogs ) = 20%
Define: Let A = male.
Let B = jogs.
The probability that a male respondent jogs is about 34%.
Write: P( A | B ) =
P( A and B )
P( A )
= Substitute 0.2 for P(A and B) and 0.58 for P(A).
0.344 Simplify.
0.2
0.58

• On Monday April 15, 1912, the RMS Titanic sank in the North Atlantic
Ocean on its maiden voyage from Southampton to New York. The
data concerned with this epoch event in history is presented in Table
Q3b. Use it to answer the following questions.

Table on Titanic Mortality
men women boys girls total
survived 332 318 29 27 706
died 1360 104 35 18 1517
total 1692 422 64 45 2223
Assuming that 1 person is randomly selected from the 2223 people aboard the Titanic,
Find P (selecting a man or a boy).
Find P (selecting a man or someone who survived).
Find P (woman or child| survivor)
Find P (survived| man)
Find P (man| survived)

Using Tree Diagrams
Jim created the tree diagram
after examining years of weather
observations in his hometown. The
diagram shows the probability of
whether a day will begin clear or
cloudy, and then the probability
of rain on days that begin clear
and cloudy.
a. Find the probability that a day will start out clear, and then will rain.
The path containing clear and rain represents days that start out clear
and then will rain.
P(clear and rain) = P(rain | clear) • P(clear)
= 0.04 • 0.28
= 0.011
The probability that a day will start out clear and then rain is about 1%.

(continued)
b. Find the probability that it will
not rain on any given day.
The paths containing clear and no rain and cloudy and no rain both represent
a day when it will not rain. Find the probability for both paths and add them.
P(clear and no rain) + P(cloudy and no rain) =
P(clear) • P(no rain | clear) + P(cloudy) • P(no rain | cloudy)
= 0.28(.96) + .72(.69)
= 0.7656
The probability that it will not rain on any given day is about 77%.

Let’s Try One
• A survey of Pleasanton Teenagers was given.
• 60% of the responders have 1 sibling; 20% have 2 or more siblings
• Of the responders with 0 siblings, 90% have their own room
• Of the respondents with 1 sibling, 20% do not have their own room
• Of the respondents with 2 siblings, 50% have their own room
Create a tree diagram and determine
A) P(own room | 0 siblings)
B) P(share room | 1 sibling)

• 60% of the responders have 1 sibling; 20% have 2 or more siblings
• Of the responders with no siblings, 90% have their own room
• Of the respondents with 1 sibling, 20% do not have their own room
• Of the respondents with 2 siblings, 50% have their own room
Create a tree diagram and determine
A) P(own room | 0 siblings)
B) P(share room | 1 sibling)

Conditional Probability (1) by using statistics.pptx

• The voters in a large city are 40 % white, 40 % black, and 20 % Hispanic. A black
mayoral candidate anticipates attracting 30 % of the white vote, 90 % of the black
vote, and 50 % of the Hispanic vote. Draw a tree diagram with probabilities for
the race (white, black, or Hispanic) and vote (for or against the candidate) of a
randomly chosen voter. What percent of the overall vote does the candidate
expect to get?
• Lactose intolerance causes difficulty digesting dairy products that contain lactose
(milk sugar). It is particularly common among people of African and Asian
ancestry. In the United States, 82 % of the population is white, 14 % is black, and
4 % is Asian. Moreover, 15 % of whites, 70 % of blacks, and 90 % of Asians are
lactose intolerant.
• Construct a tree diagram for the data given;
• What percent of the entire population is lactose intolerant?
• What percent of people who are lactose intolerant are Asian?

Question 1
• In a class of a certain school, 20 of the students study French, 12 study
Geography, and 9 study History. 12 students study only French, 4 study only
History and 5 study only Geography. Every student studies at least one of
the three subjects but no student studies all three.
a) How many students are there in the class?
b) If a student is selected at random from the class, find the probability that
the person chosen studies
i. both History and Geography
ii. exactly two subjects
c) If a student is selected at random from the list of all those who do not
study French, find the probability that the person chosen studies
i. both History and Geography
ii. only Geography

Question 2
• Suppose that A and B are events with probabilities: P(A)=1/3,
P(B)=1/4, P(A ∩ B)=1/10. Find each of the following:
1. P(A | B)
2. P(B | A)
3. P(A’ | B’)

Conditional Probability (1) by using statistics.pptx

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