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Geometry – Chapter 14 Day #4
Topic: Probability Distributions, Frequency Tables, Two Way Tables, Conditional Probabilities
Standards/Goals:
 S.CP.4.: I can use a two-way table involving categories to determine probabilities.
 S.CP.5.: I can recognize the concepts of conditional probabilities and independence in everyday
situations.
 S.CP.6.: I can calculate a conditional probability and interpret the result in the context of the given
problem.
We first want to encounter how we organize categorical data. We can do this using a frequency table:
Number of trophies Frequency
0-2
6
3-5
4
6-8
3
TOTAL:
 The frequency table above shows the number of trophies won by 13 members of a karate club.
For example: 4 members have won 3, 4, or 5 trophies.
 You can use relative frequency to compare the categories, or intervals. Relative frequency is the
ratio of the frequency of the category to the total frequency:
RELATIVE FREQUENCY = _________________________________________________________ = _____
o
o
o
The relative frequency of karate club members who won 3-5 trophies is:
Relative frequency can be used to determine probability.
The probability of randomly choosing a club member who won 6-8 trophies is:
EXAMPLE: A student counts the types of vehicles as the enter the school parking lot during a 30
minute period. She records the results in the frequency table below:
Type of Vehicle Frequency
Car
11
Van
4
Truck
8
TOTAL:
#1. How many vehicles entered the parking lot during the 30-minute period?
Find the relative frequency for each type of vehicle:
#2. Car
#3. Van
#4. Truck
#5. What is the probability of randomly choosing a van from the vehicles that entered the parking lot
during this 30-minute period?
 A probability distribution displays probabilities for all possible outcomes in a sample space.
EXAMPLE: Twelve members of the basketball team attempt 3 free throws each. The manager records
the results in the probability distribution below:
FREE THROWS MADE
0
1
2
3
FREQUENCY
1
2
6
3
PROBABILITY
1/12 2/12 6/12 3/12
#1. The sum of the frequencies is equal to the number of players:
#2. The sum of the probabilities is equal to 1:
#3. The probability that a player made 1 free throw is:
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EXAMPLE: The number of buttons on each of 10 shirts is shown in the probability distribution below:
Number of buttons
4
5
6
Frequency
1
5
Probability
1/10
5/10
#1. Complete the missing parts of the table.
#2. What is the probability that a shirt chosen at random from the set of 10 shirts has 5 buttons?
#3. What is the sum of the probabilities?
#4. What is the probability of randomly choosing a shirt that has more than 4 buttons from a set of 10
shirts?
We finally want to learn about a type of a probability called a conditional probability.
 A conditional probability is the probability that an event will occur, given that another event has
ALREADY occurred.
 You can write the conditional probability of event B, given that event A has already occurred as:
 The above is read as: “the probability of event B, given event A.”
EXAMPLE: Opinion polls:
Respondents of a poll were asked whether they were for, against or had no opinion about a bill before
the state legislature that would increase the minimum wage. What is the probability that a randomly
selected person is over 60 years old, given that the person had no opinion on the state bill?
Age Group For Against No Opinion TOTALS
18-29
310
50
20
380
30-45
200
30
10
240
45-60
120
20
30
170
Over 60
150
20
40
210
TOTALS
780
120
100
1000
#1. What is the probability that a person randomly selected has over 60 years old given that they have
no opinion?
#2. What is the probability that a randomly selected person is 30-45 years old, given that the person is
in favor of the minimum wage bill?
#3. What is the probability that a randomly selected person is NOT 18-29, given that the person is in
favor of the minimum wage bill?
#4. What is the probability that a randomly selected person is AGAINST the minimum wage bill?
#5. What is the probability that a randomly selected person is FOR the minimum wage bill, given that
they are 30-45 years old?
#6. What is the probability that a randomly selected person is 18-29 years old, given that they have no
opinion of the minimum wage bill?
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HOMEWORK – Chapter 14 Day #4
A camp counselor records the number of camp attendees who participate in the daily activities. The
results are shown in the table below:
Camp Activities
ACTIVITY
NUMBER OF PEOPLE
Waterskiing
12
Hiking
18
Canoeing
13
Find the relative frequency of each activity:
#1. Waterskiing
#2. Hiking
#3. Canoeing
A spinner has 3 equal sections colored red, blue, and green. A student conducts an experiment where
she spins the spinner twice and records the results. The results are shown in the frequency table
below:
COLORS
RR RB RG BB BR BG GG GR GB
FREQUENCY 3
5
2
1
4
2
3
2
1
#4. What is the probability of spinning red exactly once on the next two spins?
#5. What is the probability of spinning blue twice on the next two spins?
At a mini-golf course, 20 friends each take 5 shots to try to get a hole in one. The results are shown in
the probability distribution below. Complete the table.
Number of
0
1
2
3
4
5
Holes-in-one
FREQUENCY
2
6
5
4
2
1
PROBABILITY
#6. _____
#7. _____
#8. ______
#9. ______ #10. ______ #11. ____
A student records the favorite season for 50 students. The results are shown in the table below:
SEASON NUMBER OF RESPONSES
Winter
Spring
Summer
Fall
#12. What is the relative frequency of spring?
7
13
19
11
#13. What is the relative frequency of winter?
#14. If the table included the number of responses for only THREE of the seasons, could you determine
the relative frequency of the remaining seasons? Explain.
#15. CONCEPT QUESTION: How are the relative frequencies in a frequency table mathematically
related?
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#14. A student randomly chooses songs on her MP3 player. Out of 30 different choices, she chooses 8
hip-hop songs, 4 country songs, 11 rock songs, and 7 classical songs. What is the probability that the
student chooses a country song?
#15. WRITING: A certain probability distribution includes simplified fractions for some of the
probabilities. Will the sum of the numerators be equal to the total frequency? Explain.
#16. ERROR ANALYSIS: A cross country coach makes the probability distribution below for the number
of wins for some of the team members.
Number of wins 0
1
2
3
4
Frequency
7
4
6
5
3
Probability
7/10 4/10 6/10 5/10 3/10
Explain the error that the coach made when making the table.
For the following problems, using the two way frequency table below to find the probability of each
event.
Attendance at Soccer Camp
6th graders
BOYS
7
GIRLS
8
TOTALS
15
7th graders
6
7
13
8th graders TOTAL
10
23
12
27
22
50
#17. What is the probability of randomly selecting a 7th grade girl?
#18. What is the probability of randomly selecting an 8th grade boy?
#19. What is the probability of randomly selecting a 6th grade girl?
#20. What is the probability of randomly selected a girl, given that she is in the 7th grade?
#21. What is the probability of randomly selecting a 6th grader, given that they are a boy?
#22. What is the probability of randomly selecting an 8th grader, given that they are a girl?
#23. What is the probability of randomly selecting a student who is NOT an 8th grader?
#24. What is the probability of randomly selecting a student who is NOT a boy?
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