Download Statistics Chapter 5 Probability Models

Section 5.1
Constructing Models of Random Behavior
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Main Concepts:
List all possible outcomes of a chance process in a
systematic way
Design simulations and use them to estimate
probabilities
Use the Addition rule to compute probability that
event A or event B (or both) occurs
Use the Multiplication rule to compute probability
that event A and event B both occur.
Compute Conditional probabilities, the probability
that event B occurs given that event A occurs (the
most difficult)
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An event is a set of possible outcomes from a
random situation: rolling dice, drawing a
card, pulling a marble from a bag, result of
some type of spinner…
Probability is a number between 0 and 1 (or
between 0% and 100%.
 Something that is certain to occur has a
probability of 1.
 Something that will not occur has a probability of
0.
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The probability that event A occurs is
denoted by P(A).
The probability that event A does not occur is
then, P(not A) = 1 – P(A)
 “not A” is also called the compliment of A.
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If you have a list of all possible outcomes and
all outcomes are equally likely, the probability
of a specific outcome is: the number of
outcomes of that event / the total number of
equally likely outcomes
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Page 288: Identifying Tap or Bottled water:
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Probability Distribution: a chart or graph
showing the possible values from the random
process and the probabilities of each.
 The sum of the probabilities must be….?
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A Sample Space for a chance process is a
complete list of disjoint outcomes. All of the
outcomes in a sample space must have a total
probability equal to 1.
 Disjoint: two different outcomes can’t occur in
the same opportunity. (AKA: mutually exclusive)
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Think of a Tree Diagram which maps out the
possibilities.
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If you have n1 possible outcomes for stage 1
and n2 possible outcomes for stage 2, n3
possible for stage 3 and so on, then the total
possible for the stages together is n1n2n3.
 2 people guessing T or B.
 3 people guessing T or B.
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6 possibilities on each. n1=6, and n2=6
So there are 6*6 = 36 equally likely outcomes.
Notice for example 3,4 and 4,3 have the same
result, but were different in how they
happened, so they are considered two
different outcomes.
However 3,3 is the same as 3,3 so those are
not two different outcomes.
1
2
3
4
5
6
1
1,1 = 2
1,2 = 3
1,3 = 4
1,4 = 5
1,5 = 6
1,6 = 7
2
2,1 = 3
2,2 = 4
2,3 = 5
2,4 = 6
2,5 = 7
2,6 = 8
3
3,1 = 4
3,2 = 5
3,3 = 6
3,4 = 7
3,5 = 8
3,6 = 9
4
4,1 = 5
4,2 = 6
4,3 = 7
4,4 = 8
4,5 = 9
4,6 = 10
5
5,1 = 6
5,2 = 7
5,3 = 8
5,4 = 9
5,5 = 10
5,6 = 11
6
6,1 = 7
6,2 = 8
6,3 = 9
6,4 = 10
6,5 = 11
6,6 = 12
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In random sampling, the larger the sample, the closer
the proportion of successes in the sample tends to be
to the proportion in the population.
 ie: Think of flipping a coin.
 If you flipped the coin twice, would you expect Heads 50%
of the time (would you be surprised if you got two Heads)?
 If you rolled the dice 5000 times, would you expect Heads
approximately 50% of the time (would you be surprised if
you got 5000 Heads)?
 This is an example of the Law of Large Numbers.
 You are really comparing theoretical probability to
experimental probability.
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A sample space together with an assignment
of probabilities.
 Each will have probability between 0-1.
 All together they will sum to 1.
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Homework: p296 P1-P9, E1,2,3,7,9,13