Probability

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Probability
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Would you bet your life?

• Humans not only bet money when they gamble, but
  also bet their lives by engaging in unhealthy
  activities such as smoking, drinking, using
  drugs, and exceeding the speed limit when
  driving.
• Many people do not care about the risks involved
  in these activities since they do not understand
  the concepts of probability.
• On the other hand, people may fear activities
  that involve little risk to health or life
  because these activities have been
  sensationalized by the press and media.
• In this session, we will learn about
  probability.

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Probability

• Probability can be defined as the chance of an
  event occurring.
• It grew out of the use of cards, coins and dice
  in games of chance.

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Probability

• The study of probability helps us figure out the
  likelihood of something happening.
• For instance, when you roll a die, you might ask
  how likely you are to roll a five.
• In math, we call the "something happening" an
  "event."
• The probability of the occurrence of an event can
  be expressed as a fraction or a decimal from 0 to
  1.
• Events that are unlikely will have a probability
  near 0, and events that are likely to happen have
  probabilities near 1.

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Notation for Probabilities

• P - denotes a probability.
• A, B, and E - denote specific events.
• P (A) - denotes the probability of event A
  occurring.

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Probability

Range of Probabilities Rule The probability of an
event E is between 0 and 1, inclusive. That is

100
0
0 ? P(E) ? 1.
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Probability experiments

• A process (or an activity) such as flipping a
  coin, rolling a die, or drawing a card from a
  deck is called a probability experiment
  (activity).
• In other words, A probability experiment is a
  chance process that leads to well-defined results
  called outcomes.
• The result of a single trial in a probability
  experiment is the outcome.
• A trial means flipping a coin once, rolling one
  die once, or the like.
• When a coin is tossed, there are two possible
  outcomes head or tail.
• ( Note, we exclude the possibility of a coin
  landing on its edge.)
• Rolling a single die, there are six outcomes 1
  ,2 , 3 , 4, 5, 6.

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Probability experiments

• In any experiment, the set of all possible
  outcomes is called the sample space.
• Experiment (Activity) Sample Space
• Toss one coin Head,Tail (H,T)
• Roll one die 1, 2, 3, 4, 5, 6
• True/false question True, False
• Toss two coins HH, TT, HT, TH

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Example

• Find the sample space for the gender of the
  children if a family has three children. Use B
  for boy and G for girl.

Third Child
Outcomes BBB BBG BGB BGG GBB GBG
GGB GGG
First Child
Second Child
B
B G
B
G
B
G

G
B
B
G
B
G
G
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Example

• Find the sample space for rolling two dice
• Die 2
• Die 1
• 1 2 3 4
  5 6
• 1 (1,1) (1,2) (1,3) (1,4)
  (1,5) (1,6)
• 2 (2,1) (2,2) (2,3) (2,4)
  (2,5) (2,6)
• 3 (3,1) (3,2) (3,3) (3,4)
  (3,5) (3,6)
• 4 (4,1) (4,2) (4,3) (4,4)
  (4,5) (4,6)
• 5 (5,1) (5,2) (5,3) (5,4)
  (5,5) (5,6)
• 6 (6,1) (6,2) (6,3) (6,4)
  (6,5) (6,6)

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Events
An event is a set of outcomes of a probability
experiment. we call the "something happening" an
"event." An event can be one outcome or more than
one outcome.

• Example A die is rolled.
• Event E is rolling the number 6.
• a single outcome, so we can E a simple event.
• 2. Event A is rolling an even number.

The outcomes of event A are 2, 4, 6, so event A
is not a simple event.
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Types of Probabilities

• 1. Classical Probability (or Theoretical
  Probability)
• 2. Empirical Probability ( or Relative Frequency
  Approximations of Probability)
• 3. Subjective Probability

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Classical Probability

•  One event, all outcomes equally likely
• Suppose we have a jar with 4 red marbles and 6
  blue marbles, and we want to find the probability
  of drawing a red marble at random (E). In this
  case we know that all outcomes are equally
  likely any individual marble has the same chance
  of being drawn.
• To find the probability of an event E with all
  outcomes equally likely, we use a fraction
• What's a favorable outcome? In our example, where
  we want to find the probability of drawing a red
  marble at random, our favorable outcome is
  drawing a red marble.
• What's the total number of possible outcomes
  (sample space)? In our problem, the sample
  space consists of all ten marbles in the jar,
  because we are equally likely to draw any one of
  them.

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Probability of drawing a red marble

• Using our basic probability fraction, we see that
  the probability of drawing a red marble at random
  is

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Classical Probability
Classical probability is used when each outcome
in a sample space is equally likely to occur.
The classical probability for event E is given by
Example A die is rolled. Find the probability
of Event A rolling a 5.
There is one outcome in Event A 5
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Toss a coin

• What is the probability of getting a head when a
  coin is tossed?
• P( getting a head)

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Gender of Children

• If a family has three children, find the
  probability that all the children are girls.

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Expressing Probabilities

• When expressing the value of a probability,
  either
• Give the EXACT fraction or decimal (preferred)
• Round off the FINAL decimal result to three
  significant digits.
• In all cases where practical, an exact answer is
  best.
• For example is better than 0.333.

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Empirical Probability
Empirical probability is based on observations
obtained from probability experiments. The
empirical frequency of an event E is the relative
frequency of event E.
Example A travel agent determines that in every
50 reservations she makes, 12 will be for a
cruise. What is the probability that the next
reservation she makes will be for a cruise?
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Probabilities with Frequency Distributions
Example The following frequency distribution
represents the ages of 30 students in a
statistics class. What is the probability that a
student is between 26 and 33 years old?
P (age 26 to 33)
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Subjective Probability
Subjective probability results from intuition,
educated guesses, and estimates. This type of
probability is not scientific.
Example A business analyst predicts that the
probability of a certain union going on strike is
0.15.
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Law of Large Numbers
As an experiment is repeated over and over, the
empirical probability of an event approaches the
theoretical (actual) probability of the event.
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Complementary Events
The complement of Event E is the set of all
outcomes in the sample space that are not
included in event E. (Denoted .)
Example There are 5 red chips, 4 blue chips,
and 6 white chips in a basket. Find the
probability of randomly selecting a chip that is
not blue.