10'1 Introduction to Probability

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10.1 Introduction to Probability
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Theoretical Probability

• If all outcomes in a sample space are equally
  likely, then the theoretical probability of an
  event A, denoted P(A), is defined by
• P(A)

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Example 1

• Find the probability of randomly selecting a blue
  disk in one draw from a container that contains 2
  red disks, 4 blue disks, and 3 yellow disks.

4/9
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Fundamental Counting Principle

• If there are m ways that one event can occur and
  n ways that another event can occur, then there
  are mxn ways that both events can occur.
• Tree diagrams illustrate the Fundamental Counting
  Principle.

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Example 2

• A combo special at a restaurant includes a choice
  of beverage (tea, soda) and a choice of salad
  (garden, potato, and bean). How many possible
  choices for the lunch special? Make a tree
  diagram to illustrate.

2 x 3 6 possible choices
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Example 3

• How many possible passwords can be formed from 2
  letters followed by 4 digits with no numbers or
  letters excluded?

There are 26 possible letters (A-Z) and 10
possible digits (0-9)
26 x 26 x 10 x 10 x 10 x 10 6760000
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10.2 Permutations
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Permutations

• A permutation is an arrangement of objects in a
  specific order.
• The number of permutations of n objects is given
  by n factorial (written n!).
• n! n x (n-1) x (n-2) xx 2 x 1
• Example How many ways can the letters in the
  word objects be arranged?

• Answer There are 7 unique letters, so there
• are 7! ways (5,040).

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Permutations of n Objects Taken r at a Time

• Denoted nPr
• Example You have 6 disk changer in your car.
  How many ways can you listen to 6 different CDs
  if you have a selection of 10 CDs?

• Answer P(10, 6) 10P6 151,200

• Example How many 4 digit PIN numbers are
  possible if no digit can be repeated?

Answer P(10, 4) 10P4 5,040
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Permutations With Identical Objects

• The number of distinct permutations of n objects
  with rk identical objects is
• Example Find the permutations of the letters in
  the word statistics?

• Answer There are 10 letters, t is used 3 times,
  s is used 3 times, and i is used 2 times

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10.3 Combinations
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Combinations

• A combination is an arrangement of objects in
  which order is not important.
• When reading a problem you must consider whether
  order matters to know whether to use a
  permutation or combination
• Examples
• The number of ways to listen to 2 of 5 CDs
• The number of ways to purchase 2 of 5 CDs
• Its a subtle difference, but the first is a
  permutation and the second is a combination

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Combinations of n Objects Taken r at a Time

• Denoted nCr
• Example Find the number of ways to purchase 3
  different kinds of juice from a selection of 10
  different juices.

Answer Order doesnt matter. C(10, 3)
10C3 120
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Permutation or Combination?

• How many ways are there to give 3 honorable
  mention awards to a group of 8 contestants?
• How many ways are there to award first prize,
  second prize, and third prize to a group of 8
  contestants?

• Answer The order doesnt matter in the first
  (combination) but matters in the second
  (permutation).
• 1. 8C3 2. 8P3

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Combinations Example

• How many different ways are there to purchase 3
  novels and 2 non-fiction books if there are 10
  novels and 6 non-fiction books to choose from?

• Answer 10C3 x 6C2 1800

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Combinations and Probability

• In a recent survey of 25 voters, 17 favor a new
  city regulation
• and 8 oppose it.
• Find the probability that in a random sample of
  6 respondents from the survey, exactly 2 favor
  the proposed regulation and 4 oppose it.

• Find the number of outcomes in the event (17C2 X
  8C4)
• Find the number of outcomes in the sample space
  (25C6)
• Find the probability.

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Combinations and Probability

• In a recent survey of 25 voters, 17 favor a new
  city regulation
• and 8 oppose it.
• Find the probability that in a random sample of
  10 respondents from the survey, all 10 favor the
  proposed regulation.

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10.4 Using Addition With Probability
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Inclusive vs. Mutually Exclusive Events

• Inclusive events that can occur at the same
  time
• For example, rolling an even number and rolling a
  6
• Mutually Exclusive events that cannot occur at
  the same time
• For example, flipping a coin cannot be both heads
  and tails

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Probability of A or B

• Let A and B represent events in the same sample
  space.
• If A and B are mutually exclusive events, then
• P(A or B) P(A) P(B)
• If A and B are inclusive events, then
• P(A or B) P(A) P(B) P(A and B)

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Examples

• A card is drawn at random from a 52-card deck.
• Tell whether events A and B are inclusive or
  mutually
• exclusive. Then find P(A or B).
• The card is a heart or the card is an 8.
• The card is red or the card is the ace of spade.

Answer inclusive
Answer mutually exclusive
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Examples

• A card is drawn at random from a 52-card deck.
• Tell whether events A and B are inclusive or
  mutually
• exclusive. Then find P(A or B).
• The card is a number less than five or the card
  is a jack, king or queen.
• The card is not a diamond or the card is a spade.

Answer mutually exclusive
Answer inclusive
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Probability of the Complement of A

• Complement of A all the outcomes in the
• sample that are not in A
• Denoted Ac
• Example
• If A is the event rolling an even number, then Ac
  is the event rolling an odd number.
• P(A) P(Ac) 1 (100)
• P(A) 1 P(Ac)
• P(Ac) 1 P(A)

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Examples

• How many integers from 1 to 600 are divisible
• by 2 or 3?
• Even numbers are divisible by 2 (2, 4, 6, 8)
• 600/2 300 numbers
• Every third number is divisible by 3 (3,6,9,12)
• 600/3 200 numbers
• 6 is the first number that repeats in both lists
  meaning you counted multiples of 6 twice
• 600/6 100 numbers
• P(divisible by 2 or 3) 300 200 100 400
  numbers
• Find the probability that a random integer from 1
  to 600 is divisible by neither 2 or 3.
• This is the complement of the above situation.
• 600 400 200 numbers wouldnt be divisible by
  either so this probability is 33

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Testing

• Friday we will do notes for both 10.5 and 10.6
• All Juniors/some sophomores periods 1-3 will not
  be
• in class Mon-Thurs
• Chapter 10 test needs to be taken Wednesday or
  Thursday after school or Friday during class
  (schedule a time to take this test BEFORE Spring
  Break!!!)
• Monday (Quiz over 10.5, test review) - 1st
• Tuesday (Quiz over 10.5, test review) 3rd
• Wednesday (Quiz over 10.6, test review) 3rd,
  1st
• Relocation Info
• 1st - TR 14 2nd T11 3rd T25

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10.5 Independent Events10.6 Dependent Events
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10.5 Probability of Independent Events

• Independent Events the outcome of one event
  CANNOT affect the probability of another event
• Example If a marble is selected from a bag of
  marbles and replaced and then a second marble is
  selected
• If events A and B are independent
• P(A and B) P(A) x P(B)

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Examples

• Find the probability of each event
• 3 heads on 3 tosses of a coin
• toss 1 2 3
• 2 heads or 2 tails appearing in 2 tosses of a
  coin
• both heads or both tails

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10.6 Dependent Events

• Dependent Events outcome of one event
• CAN affect the probability of another event
• Example If a marble is selected from a bag of
  marbles without replacement and then a second
  marble is selected
• A bag contains 9 red marbles and 3 green marbles.
  Find the probability of selecting a red marble
  on the first draw and a green marble on the
  second draw.
• a. The first marble is replaced. b.
  The first marble is not replaced.

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10.6 Conditional Probability

• Conditional Probability probability of
• event B, given that event A has happened,
  denoted P(BA)
• P(BA) P(A and B)
• P(A)

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Examples

• Find the probability of each event for one roll
  of a dice
• 5, given that it is an odd number
• P(5odd) 1/6 33
• 3/6
• 2, given that it is less than or equal to 5
• P(25) 1/6 20
• 5/6