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, whatProbability Example
• 10 marbles in a bag, 6 blue and 4 red. • If I draw one marble at random, whatis 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, whatis 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 ofHow 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 #1-126.1.2 - 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 seepatterns.
– ‘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 mightTheoretical v Experimental
• Theoretical probability describes whatresults 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 obtainedfrom 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 aSimple 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 atornado
• 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 closerour 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 theExample
Example – Tree Diagram
• What is the probability of flipping 3Example
• 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 theprobabilities are NOT 50/50.
More Complex Example
• This method also works when theprobabilities 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 theprobabilities 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, theprobability 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 ofpeople 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?
• RangeRemember 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, 17Data 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 isfrom 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 Varationfor Data Set #2
Mean Absolute Deviation
• Now find the Mean Absolute Deviationfor 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 thesame 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 thisMy Real Estate Agent
• The mean home price in thisneighborhood 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 thanMy Real Estate Agent
• What might be a better measure thanmean 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?0 10 20 30 40 50 60 70 80 90
1st Qtr 2nd Qtr 3rd Qtr 4th Qtr
Graphs
• What kind of graph is this?0 10 20 30 40 50 60 70 80 90 100
1st Qtr 2nd Qtr 3rd Qtr 4th Qtr
Graphs
• Answers:Two Bar Graphs
0 10 20 30 40 50 60 70 80 90 1001st Qtr 2nd Qtr 3rd Qtr 4th Qtr East 0 5 10 15 20 25 30 35 40 45
Double Bar Graph
0 10 20 30 40 50 60 70 80 90 1001st Qtr 2nd Qtr 3rd Qtr 4th Qtr
East West
Two Line Graphs
0 10 20 30 40 50 60 70 80 90 1001st Qtr 2nd Qtr 3rd Qtr 4th Qtr East 0 5 10 15 20 25 30 35 40 45
Double Line Graph
0 10 20 30 40 50 60 70 80 90 1001st Qtr 2nd Qtr 3rd Qtr 4th Qtr East West
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 #56.3.2 - Circle Graphs
• Key Skill: WWBAT understand andInterpret 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 setSteps
1) Put data in orderTips 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