Transcript Section 5.1 Introduction to Probability and

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Chapter 5: Probability: What are the Chances?
Section 5.1
Randomness, Probability, and Simulation
The Practice of Statistics, 4th edition – For AP*
STARNES, YATES, MOORE
Idea of Probability
The law of large numbers says that if we observe more and more
repetitions of any chance process, the proportion of times that a
specific outcome occurs approaches a single value.
Definition:
The probability of any outcome of a chance process is a
number between 0 (never occurs) and 1(always occurs) that
describes the proportion of times the outcome would occur in a
very long series of repetitions.
Randomness, Probability, and Simulation
Chance behavior is unpredictable in the short run, but has a regular and
predictable pattern in the long run.
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 The
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A Common Question
 The
probability of tossing a coin and it
landing on Heads is 0.5. Theoretically, then,
if I toss the coin 10 times, I should get 5
Heads. However, with such a small number
of tosses there is a lot of room for variability.
 There
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are two games involving flipping a coin.
Game 1: You win if you throw 40% - 60% heads.
Game 2: You win if you throw more than 75% heads.
 For
which game would you rather toss the coin 50
times? 500 times?
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Simulations
 What
are they?
of an experiment – a way to calculate
the probability of something happening without
actually carrying out the experiment
 Imitations
 Why
 It
do we use them?
can be quicker and/or less expensive than
actually carrying out the experiment.
 The laws of probability can be confusing.
Simulation makes sense.
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Example
 Suppose
I want to know the probability that a
couple has a girl among their first four
children. Assume that:
 The
probability of having a boy =
the probability of having a girl = 0.5
 What
real life object simulates 2 outcomes of
equal likelihood?
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Example Continued

So, we could designate HEADS = having a girl.
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Simulation:
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Toss the coin 4 times and record whether HEADS shows up at least
once.
Or, we could use a well-shuffled deck of cards. Let red = having a
girl and black = having a boy. Choose 4 cards and record whether a
red card shows up at least one.
Or, we could use a table of random digits. Let even = having a girl
and odd = having a boy. Choose 4 single digits. Record whether an
even number shows up at least once.
Repeat this many times and compute the probability.
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Some more examples
 Shaq’s
forte is definitely not on the free throw
line. Let’s assume he is a 60% free throw
shooter. How could we simulate an
experiment to see how many shots he would
have to take to make 5 in a row?
 How
would this problem change if Shaq is a
47% free throw shooter?
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Synopsis of the Steps in a
Simulation (State, Plan, Do, Conclude)
 What
is the question of interest about some chance
process?
 Explicitly
detail how the simulation will be carried
out.
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What will the digits represent? What digits are not used?
How many digits will be chosen at a time from the table of
random digits?
How will you know when to stop?
What will you count?
 Perform
 Use
many repetitions of the simulation.
the results of your simulation to answer the
question of interest.
Golden Ticket Parking Lottery
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 Example:
Read the example on page 290.
What is the probability that a fair lottery would result in two
winners from the AP Statistics class?
Reading across row 139 in Table
Students
Labels
D, look at pairs of digits until you
AP Statistics Class 01-28
see two different labels from 0195. Record whether or not both
Other
29-95
winners are members of the AP
Skip numbers from 96-00
Statistics Class.
55 | 58
89 | 94
04 | 70
70 | 84
10|98|43
56 | 35
69 | 34
48 | 39
45 | 17
X|X
X|X
✓|X
X|X
✓|Sk|X
X|X
X|X
X|X
X|✓
No
No
No
No
No
No
No
No
No
19 | 12
97|51|32
58 | 13
04 | 84
51 | 44
72 | 32
18 | 19
✓|✓
Sk|X|X
X|✓
✓|X
X|X
X|X
✓|✓
X|Sk|X
Sk|✓|✓
Yes
No
No
No
No
No
Yes
No
Yes
40|00|36 00|24|28
Based on 18 repetitions of our simulation, both winners came from the AP Statistics
class 3 times, so the probability is estimated as 16.67%.
NASCAR Cards and Cereal Boxes
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 Example:
Read the example on page 291.
What is the probability that it will take 23 or more boxes to get a
full set of 5 NASCAR collectible cards?
Driver
Label
Jeff Gordon
1
Dale Earnhardt, Jr.
2
Tony Stewart
3
Danica Patrick
4
Jimmie Johnson
5
Use randInt(1,5) to simulate buying one box of
cereal and looking at which card is inside. Keep
pressing Enter until we get all five of the labels
from 1 to 5. Record the number of boxes we
had to open.
3 5 2 1 5 2 3 5 4 9 boxes
4 3 5 3 5 1 1 1 5 3 1 5 4 5 2 15 boxes
5 5 5 2 4 1 2 1 5 3 10 boxes
We never had to buy more than 22 boxes to get the full set of cards in 50 repetitions of
our simulation. Our estimate of the probability that it takes 23 or more boxes to get a
full set is roughly 0.
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Past AP Problem 2001 #3