Probability

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Probability
Chapter 8
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Probability

• Probability is a crucial part of psychology and
  provides the basis for inferential statistics.
• It allows us to figure out the probability of
  achieving certain results by chance, allowing us
  to draw valid conclusions from these results.

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Terms and Definitions

• An experiment is an act or process that leads to
  a single outcome that can not be predicted with
  certainty.
• The basic possible outcomes of an experiment are
  called sample points.
• The collection of all sample points of an
  experiment is called the sample space.

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• E.g., A six sided die is tossed. Provide the
  sample space.

1 2 3 4 5 6
The sample space here is simply a collection of
all sample points.
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• E.g., Two coins are tossed simultaneously.
  Provide the sample space.

HH TT TH HT
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• A Venn diagram is a visual method used to
  describe sample points and sample space of an
  experiment.

Die experiment
Coin experiment
HH TT TH HT

1 2 3 4 5
6
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Properties of Probability

• Probabilities are expressed as a proportion
  between 0 and 1.
• A probability of 0 means an event is certain not
  to happen, whereas a probability of 1 means an
  event is certain to happen.
• All sample points in a sample space must add to
  1.

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• Sometimes it is convenient to express a
  probability as a percentage or as the number of
  chances of obtaining an event out of 100. Simply
  multiply the probability by 100.
• Eg, the probability of tossing a coin and getting
  tails is 0.5, in other words, theres a 50
  chance of getting tails.

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• An important probability for psychology is .05.
• This level means that 5 times out of 100 the
  results are obtained by chance.
• If the probability of an event occurring by
  chance is determined to be .05 or less, we
  consider the results statistically significant.
• Probability is based on the notion of random
  sampling. The probability of an outcome can not
  be calculated if the sample is not random.

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Calculating Probability

• We determine the probability of a single event as
  follows...

p(A) Number of outcomes favoring event A
Total number of outcomes in sample
space Eg. What is the probability of rolling a 6
on a six-sided die? p(A) 1/6 0.17
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Another Example

• A bag contains 10 marbles, 3 are black and 7 are
  white. What is the probability of drawing a
  white marble?

p(A) Number of outcomes favoring event A
Total number of outcomes in sample
space
p(A) 7/10 0.7
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Compound Events

• An event may also be the combination of two or
  more events. This is known as a compound event.
• A compound event may be the union of events (A or
  B) where we must look for the probability that
  either A or B or both occur. This is denoted as
  A ? B.

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Compound Events

• When we are asked to find the probability of a
  union of events, we use the addition rule, i.e.,
  we simply add the probability of the two events
  together.

E.g., Whats the probability of drawing a heart
or a club from a deck of cards? p(H ? C) p(H)
p(C) p(H ? C) 13/52 13/52 26/52 1/2 or
.05
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Another Example

• Whats the probability of drawing a heart, spade,
  or diamond from a deck of cards.

p(H ? S ? D) p(H) p(S) p(D) p(H ? S ? D)
13/52 13/52 13/52 p(H ? S ? D) 39/52 3/4
0.75
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Compound Events

• Note, this addition rule can only be used when
  the events are mutually exclusive, i.e., they can
  not occur at the same time.
• E.g. Drawing a heart or a club, rolling a 1 or a
  2 on a single die, having blue or brown eyes.

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• However, events are often not mutually exclusive.
• E.g., Drawing a heart or a queen, rolling a 1 or
  and odd number on a single die.

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Compound Events

• Sometimes we are asked the probability that both
  events A and B occur on a single performance in
  an experiment. This is an intersection, denoted
  as A ? B.
• In such a case, we use the multiplication rule in
  which we multiply the probabilities of the two
  events.

p(A ? B) p(A)p(B)
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An Example

• What is the probability of tossing two coins and
  receiving heads on both?

p(H ? H) p(H)p(H) p(H ? H) (1/2)(1/2) p(H ?
H) 1/4 or 0.25
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Another Example

• E. g., What is the probability of rolling double
  sixes with a pair of six-sided dice?

p(6 ? 6) p(6)p(6) p(6 ? 6) (1/6)(1/6) p(6 ?
6) 1/36 0.028
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Another Example

• Whats the probability of tossing two dice and
  getting sixes, while at the same time flipping a
  coin and getting tails?

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• Note, we only use the multiplication rule if
  events are independent.
• If the outcome of event B is not influenced by
  the outcome of event A, the events are
  independent.
• Eg., tossing heads on a coin after receiving
  tails on the previous toss.
• If the outcome of event B is influenced by the
  outcome of event A, the events are dependent or
  conditional.
• Eg., Drawing two cards from a deck and getting a
  6 on the second draw.

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• If you draw a 6 on the first draw, this changes
  the probability of drawing a 6 on the second draw.

Say we get a six on the first draw...
P(six 1st) 4/52
P(six 2nd) 3/51
What if we dont get a six on the first draw...
P(not six) 48/52
P(six 2nd) 4/51
So this shows that the two events are not
independent because the probability of the second
event depends on the outcome of the first event.
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• A bag contains 10 marbles, 3 are white, 3 are
  red, and 4 are blue. Two are drawn. Whats the
  probability of getting a red marble on the second
  draw?

If you get a red first...
P(red 1st) 3/10
P(red 2nd) 2/9
If you dont get a red first...
P(not red) 7/10
P(red 2nd) 3/9
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Events That are not Mutually Exclusive

• In a case in which we are dealing with events
  that are not mutually exclusive, and we are
  looking for a union, we use the following formula.

p(A ? B) p(A) p(B) - p(A ? B)
We must subtract events that are both A and B or
we will be double counting.
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An Example

• Consider an experiment where a die is tossed
  once. Define the following events
• A roll an even number
• B roll a number less than or equal to 3
• Find p(A ? B)

p(A ? B) p(A) p(B) - p(A ? B) p(A ? B) 3/6
3/6 - 1/6 p(A ? B) 5/6
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Another Example

• E.g., What is the probability of drawing a face
  card or a club from a deck of cards?

p(F or C) p(F) p(C)
- p(F and C)
p(F or C) 12/52 13/52 - 3/52 p(F or C)
22/52 or 0.42
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Probability and the Normal Curve

• We have seen that probability theory can be
  applied to games of chance such as picking cards
  or rolling dice, however, probability theory can
  also be applied to the normal curve.
• This is far more relevent to psychologist.

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Total Area 1 This represents 100 of The data
set.
Represents those scores below the mean, i.e.,
50 of the data set.
Represent those scores above the mean, i.e., 50
of the data set.
0.5 0.5
mean
x
We know the area under the curve can be used to
represent proportions and percentages falling
below or above a certain score. But, the area
under the curve also represents probabilities.
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Total Area 1 represents the probability of
selecting someone from the data set.
Represents the probability of selecting someone
who scored below the mean.
Represents the probability of selecting someone
who scored above the mean
0.5 0.5
mean
x
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p area under a portion of the curve
total area under the curve
Since the total area under the curve is always
equal to one
p area under a portion of the curve
What statistic gives us the area under the curve?
Z-score
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• Thus far, we have used z-scores to calculate area
  under the curve. Therefore, we can use z-scores
  to calculate probability.
• E.g., On a recent stats exam, the mean was 62
  with a standard deviation of 13. Whats the
  probability of selecting at random someone who
  scored at least 75

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This is the area we want
62
75
Find the z-score corresponding to 75. Z X - X
75 - 62 1.00 S 13
Convert the z-score to area under the curve using
Table A.
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• Remember, p area under the portion of the curve
  
• This means that the area under the curve which
  you receive from Table A is the probability of
  selecting someone who has scored 75 or higher.
• According to column C for Z 1.00, area under
  the curve .1587
• Thus, the probability of selecting someone who
  has scored 75 or higher is .1587 or 16.

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Another Example

• For the same data set, what is the probability of
  selecting someone who scored between 70 and 75?

This is the area we want.
62 70 75
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• Weve already calculated the z-score (z 1.00)
  for area above 75. We can get the area below
  75 by looking at column B.

We know the total area between 75 and 62
We will find the area between 70 and 62
and subtract it from .3413
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• First we need to find the z-score for 70.
• Z X -X Z 70 - 62 .62
  
• s 13

.2324
.3413
62 70 75
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• So the probability of choosing someone at random
  with a mark between 70 and 75 is
• 0.3413 - .2324 .11
• So there is a 11 chance of choosing someone at
  random with a mark between 70 and 75.