Probability

1
Probability

• What are the chances???

2
Probability

• The probability of an event occurring
• Whats the probability of rolling a 3 with one
  die?
• Whats the probability of rolling a 5 or 6 with
  one die?
• Whats the probability that if I select a student
  at random in class, theyll be a freshman? A
  sophomore? A junior? A senior? A girl? A boy?

3
Probability of an Event

• How did you determine these probabilities?
• The probability that an event will occur
• P(Freshman)

The sum of all possible probabilities is 1
4
Probability of an Event

• Given a box with 3 white balls, 5 red balls, 8
  orange balls 7 green balls, what is
• P(drawing a white ball)?
• P(drawing a red ball)?
• P(drawing an orange ball)?
• P(drawing a green ball)?
• P(drawing a white or green ball)?

5
Find the following with your group

• Whats the probability of drawing a 4 from a deck
  of 52 cards?
• Of drawing a 7?
• Of drawing an Ace, 2 or 3?
• Of drawing a Jack of clubs?
• Whats the probability of drawing a 9 if you know
  the card is a diamond?
• Of drawing a club?
• Of drawing a 5 or a heart?

6

• What would be the probability of drawing a seven?
  
• P(Drawing a seven)
• What would be the probability of NOT drawing a
  seven?
• Probability of an event NOT occurring
• What are the possible probabilities that we could
  have? What is the largest possibility? The
  smallest?

7
Recall the previous probabilities

• P(draw a white or green ball)
• P(draw an Ace, 2 or 3)
• P(draw a 5 or a heart)

8
Gun Control

• In the state of Arizona, about 67,000 people
  legally carry guns. In 2004, there were
  approximately 5,456,453 people residing here.
• What is the probability of encountering someone
  who has a gun (legally) in Arizona?
• Explain this probability in words.
• In Texas, 1 out of 92 people legally tote a gun.
  
• How does this compare with the numbers in
  Arizona?
• By the way, in Utah the probability is 1 in 40.
  
• The number of women in Arizona who carry a gun
  13,500. Whats the probability of encountering a
  woman in Arizona who carries a gun?
• Whats the probability of an Arizonan who carries
  a gun being a woman?
• Whats the probability of an Arizonan who carries
  a gun being a man?

9
Gun Control

• Recall, 1 in 92 people in Texas legally carry
  guns. 1 in 40 people in Utah legally carry guns.
• If 236,738 people carry guns in TX, what is the
  population of Texas?
• p 21,779,896
  people
• If 57,907 people carry guns in Utah, what is this
  states population?

10
Independent Dependent Events

• What does it mean for two events to be
  independent?
• Does it matter if two events are independent?
  Will this change the probability? Explain.
• Suppose I was going to draw two cards out of a
  deck. What is the probability of drawing a jack
  and then another jack?
• What would this depend on?
• What would these probabilities be?

11
Independent Events

• Multiplication Principle
• To calculate the probability that two independent
  events will occur in a row, just multiply the two
  respective probabilities.
• Whats the probability of tossing 4 heads in a
  row?
• Examples
• Whats the probability of drawing a diamond and
  then a heart? (With replacement)
• A non-diamond and then a red ace?
• A nationwide survey found that 35 of adults in
  the US like ice cream. If two couples sit down
  for a meal, what is the probability that all four
  like ice cream?

12
Examples (Assume Independence)

• A new skin cream can cure skin infection 90 of
  the time. If five randomly selected people with
  skin infections use this cream, what is the
  probability that
• All five are cured?
• All five still have the infection?
• A box has 10 marbles in it, 6 red and 4 white.
  Suppose we draw a marble from the box, replace
  it, and then draw another. Find the probability
  that
• Both marbles are red
• Just one of the two marbles is red
• The recovery rate from a flu is 0.9. If 4 people
  have this flu, what is the probability that
• All will recover?
• Three will recover?

13
Finding Probabilities of Dependent Events

• What does it mean for events to be dependent?
• What are some examples of dependent events?
• The probability CHANGES from one event to the
  other
• Whats the probability of drawing 2 diamonds from
  a deck of cards (with no replacement)?

14
Tree Diagram to Show
Draw a Diamond
12/51
13/5212/51 4/68
13/52
Draw a Diamond
Dont Draw a Diamond
39/51
13/5239/51 13/68
Probability
Draw a Diamond
13/51
39/52
39/5213/51 13/68
Dont Draw a Diamond
38/51
Dont Draw a Diamond
39/5238/51 38/68
Sum of these probabilities should be one!
15
Dependent Events

• Finding the probability of multiple events
• The probability of getting B is now different
  because event A has occurred

16
Example

• Given the following combo of MMs 12 blue, 15
  brown, 16 yellow, 10 red, 8 green, 10 orange.
• Whats the probability of
• Drawing a yellow 4 times in a row?
• Drawing 3 yellows then an MM that is not yellow?
  
• Drawing 2 greens and then 2 blues?
• Drawing three red out of four MMs (in any
  order)? Two red?

17
Conditional Probability

• Dependent Events use conditional probability
• Whats the probability the sum of two dice tossed
  is smaller than four if we know the first die is
  a one?
• What are the possible number combos?
• (1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
• Which of these sums to less than 4?
• (1,1), (1,2)
• Whats the probability the sum is less than 4
  given that one of the die is a one? 2/6 1/3

18
Conditional Probability

• Notation
• P(sumlt 41 rolled 1st)
• A teacher gave her class two tests. 60 of the
  class passed both tests, and 80 passed the first
  test. What percent of those that passed the
  first test also passed the second?
• Write the P statement and find the probability.

19
Death Penalty Poll

• Find the probability a student selected at random
  was
• Male.
• Male, given they were against the death penalty.
• For the death penalty given the student was a
  woman.
• Female, given they were Not Sure.

20
Poker Hands

• From a 52 card deck, how many five card hands are
  possible
• That have four twos?
• That have at least two twos?
• A royal flush?
• A hand without a royal flush?
• Three 10s?
• Three of a kind, and they are 10s?
• What is the probability of getting any one of
  these hands?

21
Poker Hands

• From a 52 card deck, how many hands are possible
  that have
• Four aces?
• Two aces and two sixes?
• Not a straight flush?
• A pair?
• What is the probability of having any of these
  hands?

22
Odds

• Entering the 2007-08 Season, here were the given
  odds of each team winning the Super Bowl
• Cardinals 60-1
• Browns 100-1
• Cowboys 15-1
• Packers 50-1
• Colts 6-1
• Chiefs 30-1
• Patriots 10-1
• Giants 20-1
• What do these mean?

23
Odds

• Odds For
• Odds Against
• Finding probability given odds
• If the odds for E are a to b,
• If the odds against E are a to b,

24
Example

• The odds for a Republican victory in the next
  presidential election are 7 to 5. What is the
  probability that a Republican victory occurs?
• The odds against the Chicago Cubs winning the
  league pennant are 200 to 1. What is the
  probability that the Cubs win the pennant?

25
Review

• How many ways can 4 people be lined up?
• How many different 3 letter codes can be formed
  from the letters A,B,C,D,E,F,G if none can be
  repeated? If repetitions are allowed?
• There are 20 students in math club. How many
  ways can a subcommittee of 3 members be formed?
• How many ways can a president and a VP be
  elected? Explain.
• How many 9 letter words (real or imaginary) can
  be formed from the letters in ECONOMICS?