**7G Slides - Chapter 6**

**6.1.1 - Basic Probability **

**Life is a Probability Problem**

• Lottery • Casino • Umbrella

• Driving in the snow • Study for a test

**Probability Example**

• 10 marbles in a bag, 6 blue and 4 red.
• If I draw one marble at random, what
**Probability Example**

• 10 marbles in a bag, 6 blue and 4 red.
• If I draw one marble at random, what
is the probability that it will be blue?

6 in 10, 6/10, 0.6, 60%

• If I throw that marble away, what is

**Probability Example**

• 10 marbles in a bag, 6 blue and 4 red.
• If I draw one marble at random, what
is the probability that it will be blue?

6 in 10, 6/10, 0.6, 60%

• If I throw that marble away, what is

the probability of drawing another blue marble?

**Dice Example**

• If we roll a fair, six-sided die, what is the probability of rolling a 6?

• An odd number?

• A number greater than 4?

**Dice Example**

• If we roll a fair, six-sided die, what is the probability of rolling a 6?

1 in 6

• An odd number?

3 in 6

• A number greater than 4?

2 in 6

• A number greater than or equal to 4?

**Key Vocabulary**

• Probability measures the likelihood of an event happening

• To show a probability as a fraction:

– The numerator is the number of possible “successes” or “examined outcomes”

– The denominator is the TOTAL number of possible outcomes

• Probabilities are between 0% and 100%

**How about a pair of dice?**

• What are the possible rolls of a pair of
**How about a pair of dice?**

• What are the probabilities of each roll?
1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

**Classwork**

• Page 262-263 #1-12
**6.1.2 - Independent Events**

• Key Skill: WWBAT recognize

**Example**

• I have a fair coin with a probability of coming up heads of 50%.

**Example**

• I have a fair coin with a probability of coming up heads of 50%.

• If I flip 5 heads in a row, what is the probability of flipping heads the 6th time?

**Independence**

• The human brain ‘wants’ to see
patterns.

– ‘Seeing’ constellations in the stars is one example.

– ‘Seeing’ numbers about to come up at a Roulette table is another.

**Key Vocabulary**

• Independent events do not affect each other.

**Dependent Events?**

• Any ideas about events that might
**Theoretical v Experimental **

• Theoretical probability describes what
results *should* look like

– Heads should come up 50% of the time

• Experimental probability describes
what results *actually* look like

**Experiment**

• Student pairs will flip a coin 20 times, with one student flipping and the other recording the results.

**Law of Large Numbers**

• The average of the results obtained
from a large number of trials should be close to the expected value, and will

tend to become closer as more trials are performed.

**Classwork**

**Simulations**

• Key Skill: WWBAT use a simulation to

**Key Vocabulary**

• Simulation is an imitation of a
**Simple Simulation**

• Say we wanted a simulation of a coin flip. • We could use a random number generator to

give us a string of ones and twos to represent heads and tails.

• Assume a ‘1’ indicates heads and a ‘2’ indicates tails

**Dice Simulation**

• We can change the simulator to generate numbers from 1 to 6 to simulate a dice game.

• Assume we roll two dice and need at least a total of 9 to win a game. What is the theoretical probability that we get at least 9?

**More Complex Simulation**

• A hospital needs Type A blood during a
tornado

• If 40% of donors are Type A, what is the probability that at least one out of 4

donors in the building is Type A?

**Type A Blood Simulation**

• We generate numbers from 1 to 10,
with 1 to 4 being Type A and 5 to 10 being other types.

• If we generate 100 sets of 4 columns, with each column a separate situation, we can calculate the probability.

**How Many Simulations?**

• The more simulations we run, the closer
our experimental probability should be to our theoretical probability.

• If we are physically flipping coins or

rolling dice, there is a trade-off between accuracy and time.

**Classwork**

• Design an experiment using a random number generator for this simulation:

– The Red Sox needs to win their final 3

games to make the playoffs. Based on the opponents, the coach determines the team has a 70% chance of winning the 1st game, an 80% chance in the 2nd game, but only a 30% chance in the 3rd game.

**6.1.3 - Fair Games**

• Key Skill: WWBAT determine if a game has an equal probability for each

**Example**

• 3 fair disks (50/50 probability for each side)

2 are yellow on both sides

1 is yellow on one side, red on the other

• One team gets 1 point if we flip the disks and they come up all yellow

• The other team gets 2 points if they are not all yellow.

**Classwork**

• Pages 268-269 #3-5, 7-10

• Work in groups of 4, play 9 rounds of

*1-2-3 Show* as described on middle of

**Monty Hall Problem**

**Monty Hall Problem**

• Form groups of 3:
– One host, one contestant, one recorder

• Play 9 rounds with the contestant

ALWAYS choosing to stay with 1st choice • Play 9 rounds with the contestant

**Compound Probability**

• Key Skill: WWBAT calculate the
**Example**

**Example – Tree Diagram**

• What is the probability of flipping 3
**Example**

• The tree diagram shows us all the

possible outcomes for flipping a coin three times. We can then count the outcomes we want and calculate the probability.

**Algebraic Method**

• To calculate compound probabilities, we MULTIPLY the individual

probabilities.

**Example**

• Assuming the chance of having a boy or a girls baby is 50/50, what is the

**Example**

• Assuming the chance of having a boy or a girls baby is 50/50, what is the

probability of having 5 girls in a row?

**More Complex Example**

• This method also works when the
probabilities are NOT 50/50.

**More Complex Example**

• This method also works when the
probabilities are NOT 50/50.

• For example, what is the probability of rolling a fair, six-sided die and getting a five, twice in a row?

**Another Example**

• The process works even when the
probabilities are different.

**Another Example**

• A basketball team needs to win its last three games to make the playoffs. The coach thinks the probabilities of them winning each game are: 70%, 80%, 30%. What is the probability they make the playoffs?

**Probability Notation**

• If we have a six-sided die, the
probability of rolling a 3 would be written as: P(3)

• The probability of choosing a red M&M from a bag would be: P(red)

**6.2.1 - Samples & Predictions**

**Key Vocabulary**

• **Population is the entire group of**

people or things being examined.

**What’s in the bag?**

• 30 colored chips are in the bag
• We will take a sample of 6 chips

**What’s in the bag?**

• 30 colored chips are in the bag
• We will take a sample of 6 chips

• Based on our sample, how many of each color are in the bag?

• Now we take a 2nd sample of 6 chips? • Now how many of each color are in the

**Sample Sizes**

• The size of a sample is a trade-off between cost/time and accuracy.

• The U.S. census is relatively accurate, but cost approximately $11 billion.

• A voter poll before an election is

**Margin of Error**

• Measures the likely difference between the SAMPLE and the actual

POPULATION.

**6.2.3 - Representative Samples**

• Key Skill: WWBAT determine if a sample is representative of a

**Problems with Early Polls**

• Conducted by phone
• Done by magazines of their subscribers

**Problems with Early Polls**

• Conducted by phone
– Only upscale families initially had phones

• Done by magazines of their subscribers

– Magazine subscribers can reflect biases

• Done in particular geographic areas

**Key Vocabulary**

• A **representative sample** is one which
has approximately the same proportion
of characteristics as the entire

population.

• Example: In an election poll, your

**Mean Absolute **

**Deviation**

• Key Skill: WWBAT define and calculate the mean absolute deviation of data

**Remember These?**

• Range
**Remember These?**

• Range is the distance from the smallest to largest value in a dataset

• Mode is the most popular data point • Mean is the average of the dataset

– Add the data points, then divide by the number of data points

**Example**

• Find the range, mode, mean and median of the following dataset:

**Example**

• Find the range, mode, mean and median of the following dataset:

3, 5, 5, 6, 9, 9, 12, 15

– Range = 12

– Mode = 5 and 9 (bimodal dataset) – Mean = 8

**Data Sets**

• Data Set #1: 6, 9, 10, 12, 13
• Data Set #2: 1, 4, 12, 16, 17
**Data Sets**

• Data Set #1: 6, 9, 10, 12, 13
• Data Set #2: 1, 4, 12, 16, 17
• Both data sets have a mean of 10, but they differ in how spread out the data points are.

**Mean Absolute Deviation**

• Measure how far each data point is
from the mean

– Because it is a distance, the numbers will always be positive

Set #1 6 9 10 12 13

Mean 10 10 10 10 10

**Mean Absolute Deviation**

• Next, find the mean of the distances:
– Distances: 4 + 1 + 0 + 2 + 3 = 10

– There are 5 data points, so the mean is 10/5 = 2

**Mean Absolute Variation**

• Now find the Mean Absolute Varation
for Data Set #2

**Mean Absolute Deviation**

• Now find the Mean Absolute Deviation
for Data Set #2

• Data Set #2: 1, 4, 12, 16, 17

Set #2 1 4 12 16 17

Mean 10 10 10 10 10

**Mean Absolute Deviation**

• Next, find the mean of the distances:
– Distances: 9 + 6 + 2 + 6 + 7 = 30

– There are 5 data points, so the mean is 30/5 = 6

**Mean Absolute Deviation**

• We can use this measure to describe a
difference between the two data sets.
• Though the two data sets have the
same mean, the second data set has a much higher “mean absolute deviation” because its data points show a much

**Key Vocabulary**

• Mean Absolute Deviation shows the average distance of data points from the mean of the data set.

**Classwork**

• Find the mean absolute deviation of the following set of test scores:

**Classwork**

• Find the mean absolute deviation of the following set of test scores:

64, 73, 84, 84, 88, 91, 96, 100

– The mean of the dataset is 85

– The total of the distances from the mean is 70

**6.3.4 - Misleading **

**Graphs and Statistics**

**My Real Estate Agent**

• The mean home price in this
**My Real Estate Agent**

• The mean home price in this
neighborhood is $500,000, so this house is a bargain at $475,000!

• Turns out there were 24 homes worth $375,000 and one mansion on a hill

worth 3,500,000!

**My Real Estate Agent**

• What might be a better measure than
**My Real Estate Agent**

• What might be a better measure than
mean for home prices?

– Home prices are usually quoted using the
**median, not the mean. **

**Percents**

• If the rate of annual inflation has

**Percents**

• If the rate of annual inflation has

moved from 1% to 2%, how much has it risen?

– One could argue that it has risen only 1 point

**6.3.1 - Double Bar and **

**Double Line Graphs**

**Graphs**

• What kind of graph is this?
**Graphs**

• What kind of graph is this?
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**Tips on Making Double Graphs**

• Choose a scale that makes sense for BOTH sets of data.

• Differentiate each data set by using a different color or shading for each set of bars/lines.

**Classwork**

• Page 296 #5
**6.3.2 - Circle Graphs**

• Key Skill: WWBAT understand and
**Interpret the Circle Graph**

• What does the graph tell us? • What does it NOT tell us?

**Interpret the Circle Graph**

• What does the graph tell us? • What does it NOT tell us?

**Interpret the Circle Graph**

**Classwork**

• Create a circle graph with protractors

**6.3.3 - Stem-and-Leaf Plots**

**Unwieldy Data Sets**

• Some data sets do NOT lend themselves to the graphs we’ve learned so far.

• What if we want to show ALL of the

**Example**

**Example**

• How do we find the range, mode, median and mean?

**Creating a Stem-and-Leaf**

• Turn to page 302 and view the data set
**Steps**

1) Put data in order
**Tips for Stem-and-Leaf Plots**

• Think carefully before choosing the scale to the left of the vertical line.

– The scale must make sense for ALL the data points