Probability and Simulation

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Chapter 6

• Probability and Simulation-
• The Study of Randomness

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Probabilityis

• The study of chance outcomes
• Simulations are used to represent real-life
  situations.
• What are the chances of a couple having a girl
  among their first 4 children?
• Use a coin to simulate it- HEADSgirls TAILSboys
• Flip the coin 4 times. Write down the of heads
  you got and divide it by 4.Thats your
  probability!
• SIMULATION is the imitation of chance behavior,
  based on a model that accurately reflects the
  phenomenon.

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Steps to Simulation

• 1. State the problem or the phenomenon.
• 2. State the assumptions.
• 3. Assign digits to represent outcomes.
• 4. Simulate many repetitions.
• 5. State your conclusions.

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Simulation Ex Tossing a Coin

• Step 1Toss a coin 10 times. What is the
  likelihood of a run of 3 heads or a run of 3
  tails?
• Step 2 Assumptions
• Head or tail is equally likely to occur.
• Tosses are independent of each other.
• Step 3 Assign digits to represent outcomes
• One digit simulates the toss of a coin
• ODD digits are heads, EVEN digits are tails
• Step 4 Simulate many reps
• Use the Table of Random digits (lets use line
  101), and group by 10s, since we are tossing a
  coin 10 times.
• Look at many groups of 10s. (25 groups of 10s)
• Step 5 State your conclusions

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Simulations with Random Digits

• Step 1 Choose a person at random from a group of
  which 70 are employed.
• Step 2 State assumptions
• One digit will represent one person
• Step 3 Assign digits
• 0,1,2,3,4,5,6employed
• 7,8,9unemployed

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Simulations with Random Digits

• Step 1 Choose one person at random from a group
  of which 73 are employed.
• Step 2 State assumptions
• 2 digits 1 person
• Selections are independent of each other, if
  replacement occurs.
• Step 3 Assign Digits
• 00,01,.72 employed
• 70, 71,., 99 not employed

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Example 6.5 Frozen Yogurt Sales

• Orders of frozen yogurt flavors have the
  following relative frequencies 38 chocolate,
  42 vanilla, and 20 strawberry. We want to
  simulate customers entering the store and
  ordering yogurt.
• Step 1 State the problem
• How would you simulate 10 frozen yogurt sales
  based on this recent history?
• Step 2 Assumptions
• 38 chocolate-42 vanilla-20 strawberry
• we assume that customers choices of flavors do
  no influence one another.

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Example 6.5 Frozen Yogurt Sales

• Step 3 Assign Digits
• Each flavor is made up of 2 digits
• 00-37 chocolate outcome
• 38-79 vanilla outcome
• 80-99 strawberry Outcome
• Step 4 Simulate many reps separate s into 2s
  and do this 10 times.
• Go to line 112 in Table B, and group digits by
  2s
• 59 63 68 88 04 04 63 47
  11 97
• V V V S C C V V
  C S
• Step 5 Conclusion
• We estimate the probability of choosing
  C3/1030, V5/1050, S2/1020. More
  repetitions would be needed to be more
  accurate!!!!

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Example 6.6 A girl or 4??? (pg. 396)

• A couple plans to have children until they have a
  girl or until they have 4 children whichever
  comes first. We will show how the use a Random
  Digits to estimate the likelihood that they will
  have a girl.
• Step 2 State assumptions
• Likelihood of boy or girl is same- .5
• Successes of children are independent of each
  other
• Step 3 Assign Digits
• 0,1,2,3,4 girl
• 5,6,7,8,9boy
• Step 4 Use the Table of Random Digits

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Example 6.6 A girl or 4?

• To read the table of random digits, read Table B
  until the couple has either a girl or 4 children.
• Use Line 130 in Table B
• 69051 64817 87174 09517 84534 06489 87201
• In these 14 repetitions, a girl was born 13
  times. Our estimate of the probability that this
  strategy will produce a girl is 13/140.93 or 93
  of the time!

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HOMEWORK

• 6.2, 6.3, 6.4, 6.5

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Simulations with a Calculator

• The RANDINT command
• MATH-PRB-RANDINT (lower bound, upper bound, of
  digits generated)
• Ex generate 5 random numbers from 0-9
• RANDINT (0,9,5)
• If you want to store your results, then do this
• RANDINT (0,9,5) STO?L1.

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Simulations with a Calculator

• If you know data is approximately normal, a mean
  and SD have been given, and you are asked to
  approximate, do this
• MATH-PRB- RandNorm(mean, sd, simulations)

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Example6.8 Gymnastics

• Carly is an Olympic gold medalist. Suppose we
  want to use 100 of Carlys previous scores from
  gymnastics meets to predict how she will do (or
  did) at the 2004 Olympics in Athens. Assume her
  last 100 performances are approximately normal
  and that her Olympic performance will follow the
  same pattern.

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Example 6.8 Gymnastics

• Lets simulate her all-around performance,
  which her scores totaled.
• Simulate each event separately, then add them!
• randNorm(9.314, 0.216, 100)?L1
• randNorm(9.553, 0.122, 100)?L2
• randNorm(9.461, 0.203, 100)?L3
• randNorm(9.543, 0.099, 100)?L4
• (L1 L2L3L4)?L5
• Now look at the mean, SD of L5 (her total
  scores)- STAT-CALC- 1-var STATS
• Using the 68-95-99.7 rule, we would expect about
  95 of Carlys scores to be between 37.44 and
  38.268.

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HOMEWORK

• 6.7, 6.15, 6.16, 6.19