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Statistics and Probability
Understand Random Samples
Random Sample
 every object or event has to have an equal
chance of being selected.
 will be representative of the entire
population.
Examples
- Using a pattern, such as selecting every
fourth person to enter the cafeteria
- Using a method such as drawing names
out of a hat, where everyone has an equal
chance
- Divide the population into groups, such as
grade level, and select randomly from
each group
Biased Sample
 does not represent the whole population
Examples
- Letting some people volunteer to take a
survey
- Choosing people who are easy to reach
- Choosing people as a group, such as
students on the honor roll.
You can use proportions to make inferences, or predictions, about populations based on sample data.
 Any inference on a population must be based on a smaller representative sample.
 The larger the sample, the more reliable the results
 Gathering data on several random samples of the same size is a good way to make reliable inferences
about a population.
Example:
Tammy was running for class president. She wanted to know if she had a good chance of winning the
election, so she decided to have her friends do a survey. Jonathan surveyed a random sample of 10
students from the school. 7 of them said they would vote for Tammy. If there are 230 students in the
school, and all of them vote, what is Jonathan’s bet estimate of the number of students who would vote
for Tammy?
Solution 1:
Proportion
Solution 2:
7
𝑥
7
=
= 70%
10
230
10
70% of 230
10x = 7(230)
0.7 x 230
10x = 1610
=161 students
x = 161 students
Probability Concepts
Probability is the measure of how likely an
event is to occur.
Event: a situation or experiment that can have
many different results, or outcomes.
Probability is expressed as a number between 0
and 1.
Theoretical Probability:
- the measure how out likely an event is to occur when
each outcome is equally likely.
- What you would expect to happen
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
P(event) = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
A fair coin is flipped once. It can land
either heads-side up or tails side up.
The theoretical probability of landing
on heads is ½
A fair coin is flipped 100 times. The
coin lands on heads 45 of those times.
The experimental probability of
45
9
landing on heads is 100 𝑜𝑟 20
Experimental Probability: based on repeated trials from an
experiment or observation.
- Outcomes in experimental probability are not equally
likely.
- You can make predictions for the likelihood of future
events based on results of experimental probability
(This will give a reasonable estimate not an exact amount.)
P (event) =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
Sample Space: the set of all possible outcomes for an experiment
Trial: an event from an experiement
Probability of Compound Events
Compound Event – consists of two or more events
- you can use Lists, Tables, or Tree Diagrams
Tree Diagram: visual representation of outcomes using
branches.
Example: Use a tree diagram to show the possible
outcomes of flipping a coin three times. 
What is the probability that the coin will land heads up
exactly once?
There are 8 possible outcomes.
Heads results exactly once in 3 of the possible outcomes.
3
So, the probability that heads results exactly once is: P = 8
You can also find the probability of
compound events by multiplying the
probability of each separate event.
1
Rolling a 2 once: P= 6
(one 2 out of 6 values)
1
1
1
Rolling a 2 twice: P = 6 x 6 = 36
You TRY:
1) Are these methods for selecting a sample fair or biased. Why?
a. The events committee wants to survey students about a school dance. The committee is
meeting the gym, where the girl’s basketball team is practicing. They survey the players on
the girls’ basketball team.
b. A store owner wants to survey customers about the products he sells. He programs the
computer to select 100 costumers from the mailing list and sends them each a survey.
2) Thomas wants to estimate how many of the 534 students in his school owned cell phones. He took
a random sample of 25 students. In his sample, 15 students said they owned cell phones. What is
the best estimate for the number of students in the school who owns cell phones?
a. 320 b. 509
c. 890
d. 15
3) A spinner has four sections: a tree, a flower, a cloud, and a sun. Jordan spins the pointer on the
spinner 20 times. The pointer lands on the tree 3 times, the flower 9 times, the cloud 2 times, and
the sun 6 times. Based on the experiment, what is the probability the spinner lands on the sun?
4) Mindie tosses a coin 50 times and gets 32 heads. Adam tosses the same coin 50 times and gets 28
heads. Cam tosses the coin 50 times and gets 21 heads. Jade tosses the coin 50 times and gets 24
heads. Use the results to predict the number of heads in 1,000 tosses.
5) Reese and James are playing rock, paper, scissors.
a.
b.
c.
d.
Draw a tree diagram to show the possible outcomes
How many possible outcomes are there?
What is the probability of at least 1 rock coming up?
What is the probability of 2 rocks coming up?
6) At a smoothie stand, you can order a fruit smoothie in three different sizes: small, medium, and
large. You can choose strawberries, mango, or both fruits. Then, you can choose to add yogurt or
not to. What is the probability that someone will order a smoothie with strawberries and yogurt in
it?
7) A seventh grader has 2 siblings. Model the possible genders of a family with three children.
a) There are ______ possible outcomes.
b) What is the probability that this seventh grader is a girl with a brother and sister?
c) What is the probability that there are at least 2 girls?
d) Name an outcome with a probability of 1/8
8) Andre flips a fair coin 4 times. What is the probability that exactly 2 flips wil results in heads and 2
flips will result in tails?
9) A spinner is divided into 12 equal sections and numbered 1-12. Emmanuel spins the arrow on the
spinner. What is the probability that the arrow lands on a number less than 5?
10)Mary recorded the number of days it was sunny last month. In the past 30 days, it was sunny 80%
of the time. Mary concludes that it will be sunny 80% of the year. Is this conclusion reasonable?
Explain.
11) Look at the team data in the table below.
Team A
Team B
72
75
73
70
69
69
70
67
71
70
74
68
Part A: What is the mean absolute deviation for Team A:
Part B: What is the mean absolute deviation for Team B:
Part C: What team has a greater measure of variability? How do you know?
12)There is a 50% chance that it will snow each of the next 3 days. Fay wants to know the probability
it will snow all three days.
Part A: Explain how a coin can help you determine the probability.
Part B: Perform a simulation or draw a tree diagram to find the probability that it will snow all
three days.
Extra Practice:
1. Jung wanted to find out which band was most popular among the high school students. Where should
be conduct his survey to get the most unbiased survey?
a. in the school cafeteria
b. at a school dance where band A is playing
c. at the record store where a member of band B is giving autographs
d. at the next school board meeting
2. Feticia wanted to find out which hip hop video was most popular among the junior high school
students. Where should she conduct her survey to get the most unbiased survey?
a. At a teacher’s meeting
b. In her physical education class where her classmates all like jazz
c. In the band hall where hip-hop music is not allowed
d. At a school dance
3. Earl wanted to find out which sport was most popular among high school students. Where should he
conduct his survey to get the most unbiased survey?
a. At a game for sport A
b. At the tryouts for sport B
c. At a chess club meeting
d. At physical education sign-ups
4. A track coach wanted to know how fast an average 7th grade girl at her school could run a 400-meter
dash. There are 55 seventh graders at her school. How could the track coach find a reasonable
estimate for the average 400-meter time?
a. By asking one of the 7th grade girls how fast she could run
b. By timing one 7th grade girl and using that time
c. By timing two 7th grade girls and averaging their times
d. By timing twenty 7th grade girls and averaging their times
5. Jared checked the school soda vending machine sales each hour between 8 am and 3 pm. The 11 am
count each day averaged about 20 sodas. Jared then predicted about 20 sodas will be sold each hour
of the day. What is wrong with this prediction?
a. Students might buy pizza
b. It might start raining
c. More sodas are usually sold during lunchtime
d. Students may not have any more money
6. To show that M&M candy bags have more of a certain color than other colors you could:
a. Survey your friends for their favorite M&M color
b. Sort several bags and calculate percentages
c. Count only the bag you just bought
d. Tell people to buy more M&Ms
7. Peter wants to know the type of music the students at his school prefer for the homecoming dance.
Which method of survey would best represent the entire student body?
a. Randomly survey the teachers
b. Randomly survey 2 students from each homeroom
c. Survey the student council officers of each class
d. Randomly survey 50 sophomores
8. The PTA at Bill Moore Junior High said that they will request the school administration to change the
school’s mascot if at least 75 of the 110 PTA members approve it. You interviewed a random sample
of 32 members and found that 23 of them approve it. Based on your survey, do you think enough
parents will approve the decision?
a. No, in the present survey sample, everyone did not approve it
b. Cannot decide because the sample does not include all the parents
c. Yes, because everyone likes a change
d. Yes, based on the sample, almost 80 parents will approve it
9. There are 1,000 jellybeans in a large box. Bob took a scoop of jellybeans from the box. He counted 15
purple jellybeans, 20 yellow jellybeans, and 5 black jellybeans. If the scoop was a good sample of the
jellybeans in the large box, how many yellow jellybeans would you expect to find the large box?
a. 125 yellow jellybeans
c. 375 yellow jellybeans
b. 200 yellow jellybeans
d. 500 yellow jellybeans
10. In a random survey of teenagers, seventeen out of twenty teens said they eat breakfast every
morning. What is a reasonable prediction for the number of teens out of 1,280 who eat breakfast
every morning?
a. 340
c. 1,088
b. 940
d. 1,260
11. A survey showed that 70% of students would select roller coasters as their favorite ride at an
amusement park. Out of 5,000 students, predict how many would select roller coasters as their
favorite ride?
a. 3,500
c. 350
b. 1,500
d. 150
12. A random survey found that 5 out of 6 people in a community visit a dentist on a regular basis. If there
are 4, 320 people in the community, what is a reasonable prediction for the number of people who
would visit a dentist on a regular basis?
a. 720
c. 2,880
b. 864
d. 3,600
13. A survey found that 6 out of 8 students own a pet. If there are 320 students in a school, what is
reasonable prediction for the number of students who own pet?
a. 240
c. 24
b. 80
d. 8
14.) Sawyer has a drawer of mittens. 6 mittens are blue, 4 mittens are black, and 2 mittens are
red. What is the probability that Sawyer randomly pulls out a pair of black mittens?
a.
1
3
b.
1
9
c.
1
10
d.
4
33
15.) Joe and Jill take turns spinning this spinner. Joe gets appoint if the arrow lands on an even
number and Jill gets a point if the arrow lands on an odd number. Is this a fair game?
a.
b.
c.
d.
Yes, because there are 8 choices.
Yes, because the outcomes are equally likely.
No, because there is a greater probability of spinning an even number.
No, because there is a greater probability of spinning an odd number.
16.) What is the probability of spinning “red” or “green” twice in a row on the spinner below?
a.
9
64
b.
3
32
c.
3
8
d.
4
64
17.) Avery flips a quarter 50 times and recorded the results in the table below:
Heads
Tails
IIII IIII
IIII IIII
IIII IIII
IIII IIII
IIII III
II
How many more times would Avery have expected to land on tails according to theoretical
probability?
a. 25 more times
b. 5 more times
c. 8 more times
d. 3 more times