PROBABILITY

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PROBABILITY

• Chapter 9

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Section 9-1

• Review Percents and Probability

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Experiment

• An activity that is used to produce data that can
  be observed and recorded
• Example rolling a die
• Example - tossing a coin
• Example drawing a card

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Outcome

• The result of each trial of an experiment.

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Event

• Any one of the possible outcomes or combination
  of possible outcomes of an experiment

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Experimental Probability

• Represents an estimate of the likelihood of an
  event, E, or desired outcome
• P(E) of observations of E
• total of observations

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Theoretical Probability

• P(E) of favorable outcomes
• of possible outcomes

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Sample Space

• The set of all possible outcomes of the
  experiment
• Tossing a coin
• S H, T
• Rolling a dice
• S 1, 2, 3, 4, 5, 6

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Tree Diagram

• A diagram that lists one part of an event and
  then adds branches to show all the outcomes
  involving that part of the event

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Example

• In an experiment, a coin is tossed and a number
  cube is rolled.
• Make a tree diagram beginning with the possible
  outcomes of the coin toss

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Relative Frequency

• Compares the number of times the outcome occurs
  to the total number of observations

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Example

• The more often you toss a coin, the closer you
  will come to tossing an equal number of heads and
  tails.

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Section 9-2

• Problem Solving Skills Simulations

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Section 9-3

• Compound Events

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Compound Event

• Made up of two or more simpler events
• Probability of a compound event is the
  probability of one event and/or another occurring

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Probability

• The probability of a compound event is
  represented by P(A n B)
• The probability of one event or another occurring
  is written P(A ? B)

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MUTUALLY EXCLUSIVE EVENTS

• Events that cannot occur at the same time
• Example A die is rolled. The events, getting
  an even number and getting an odd number are
  mutually exclusive.

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MUTUALLY EXCLUSIVE EVENTS

• If two events A and B are mutually exclusive then
• A ? B Ø
• and

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Mutually Exclusive Events

• For mutually exclusive events only
• P(A ? B) P(A) P(B)

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EXAMPLE MUTUALLY EXCLUSIVE EVENTS

• Suppose a die is tossed.
• Let A be the event that an even number turns up
• Let B be the event that an odd number turns up,
  then

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Mutually Exclusive Events

• A 2, 4, 6, and B 1, 3, 5
• A ? B Ø

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THEOREM

• If A and B are not mutually exclusive events,
  then
• P(A ? B)
• P(A) P(B) P(A ? B)

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Example -

• A card is drawn at random from a deck of 52
  playing cards. Find the probability that the
  card is a heart or an ace.
• A card is a heart
• B card is an ace
• P(A ? B) P(A) P(B) P(A ? B)

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Section 9-4

• Independent and Dependent Events

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INDEPENDENT EVENTS

• Two events are independent if the result of the
  second event is not affected by the result of the
  first event.

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INDEPENDENT EVENTS

• The events A and B are independent if, and only
  if
• P(A ? B) P(A) P(B)

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Example

• A bag contains 3 red marbles, 4 green marbles and
  5 blue marbles. One marble is taken at random
  and then replaced. Then another marble is taken
  at random.
• Find the probability that the 1st marble is red
  and the 2nd is blue.

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DEPENDENT EVENTS

• Two events are dependent if the result of one
  event is affected by the result of another event

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DEPENDENT EVENTS

• The result of event A affects event B
• P(A ? B)
• P(A) P(B, given that A occurred)
• P(A) P(BA)

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Example

• A bag contains 3 red marbles, 4 green marbles and
  5 blue marbles. One marble is taken at random
  and is not replaced. Then another marble is
  taken at random.
• Find the probability that the 1st marble is red
  and the 2nd is blue.

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Section 9-5

• Permutations and Combinations

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Fundamental Counting Principle

• If there are two or more stages of an activity,
  the total number of possible outcomes is the
  product of the number of possible outcomes for
  each stage

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Example

• At a pizza place there are three sizes (Large,
  Medium, and Small). There are also five choices
  of toppings (cheese, pepperoni, sausage, onions,
  peppers).

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• How many different pizzas with one topping could
  a customer order?
• What is the probability that a customer will
  order a Medium pizza with sausage?

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Example

• A store sells shirts in 8 sizes. For each size,
  there is a choice of 5 colors. For each color,
  there is a choice of 6 patterns. How many
  different shirts does the store have?

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• What is the probability that a customer will buy
  a large shirt that is blue with stripes?

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PERMUTATION

• An arrangement of items in a particular order.
• n! (n factorial)
• n(n-1)(n-2)(2)(1)

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FACTORIAL

• 5! 5 x 4 x 3 x 2 x 1
• 0! 1

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EXAMPLE PERMUTATIONS

• How many different ways can the letters a, b,
  and c be arranged if all the letters are used?
• 3!
• (a,b,c), (a,c,b), (b,c,a), (b,a,c), (c,a,b),
  (c,b,a)

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PERMUTATIONSNO REPETITIONS

• Uses only a part of the set without repetitions
• nPr n!__
• (n-r)!
• n number of items
• r number of items taken at a time

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EXAMPLE PERMUTATIONS

• How many different ways can the letters a, b,
  c, and d be arranged if only three different
  letters are used?
• 4!__
• (4 - 3)!

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ANSWER

• How many different ways can the letters a, b,
  c, and d be arranged if only three different
  letters are used?
• 4! 24

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COMBINATION

• An arrangement of items in which order is not
  important.
• nCr n!__
• (n-r)!r!

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COMBINATION

• nCr n!__
• (n-r)!r!
• n number of different items
• r number of items taken at a time

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EXAMPLE COMBINATIONS

• How many different ways can a 2-person committee
  be chosen from 8 people if there are no
  restrictions?
• 8!____
• (8 - 2)!2!

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EXAMPLE COMBINATIONS

• A random drawing is held to determine which 2 of
  the 6 members of the math club will be sent to a
  regional math contest.
• How many different pairs of two could be sent to
  the contest?

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EXAMPLE COMBINATIONS

• How many combinations of three letters could you
  make out of the letters a, b, c, d, e, and f?

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EXAMPLE COMBINATIONS

• A popular touring band has 20 songs. How many
  combinations of songs can the band play in their
  opening 3-song set?

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Section 9-6

• Scatter Plots and Boxplots

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SCATTER PLOT

• A type of visual display showing a relationship
  between two sets of data, represented by
  unconnected points on a grid.

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BOX-AND-WHISKER PLOT

• A type of visual display showing how data are
  dispersed around a median. It does not show
  specific items in the data.
• but

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BOX-AND-WHISKER PLOT

• It shows the median and the extremes of a set of
  data. The lower half of the data, called the
  lower quartile, and the median of the upper half
  called the upper quartile.

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• END