Probability

1
Chapter 5

• Probability

2
Trials

• Each time that we determine whether or not an
  outcome has occurred, we have performed a trial.
• Common trials that are used as examples are
  flipping coins, drawing cards, and answering
  multiple choice questions.

3
Successes

• A success occurs when the outcome of a trial is
  the specific event we are interested in.
• Successes are not necessarily positive events. If
  we are studying infant mortality rates, then a
  success occurs when a baby dies.

4
Probability

• Probability is the proportion of trials that will
  result in a success.

5
Classical Probability

• Classical probability can be calculated from our
  knowledge of the structure of the trials, before
  any experiments have actually taken place.

6
Empirical Probability

• If we do not have enough information to calculate
  the probability of an event, then we perform a
  series of trials, and record the number of
  success observed. The estimated probability of
  the event is the ratio of the number of successes
  observed to the number of trials performed.

7
Probability

• Probability must always be a number between 0 and
  1, inclusive.
• How can a negative number of successes occur?
• How can there be more successes than trials?
• If the probability of an event is 0, then it is
  impossible it will never occur.
• If the probability of an event is 1, then it will
  always occur.

8
Tree Diagram

• Illustrates outcomes by diagramming possible
  choices as branches of a tree.
• Tree diagrams are good for illustrating why the
  counting techniques that we will be studying
  work.
• Tree diagrams are very cumbersome, and become too
  large to be reasonable after only a few choices.

9
Fundamental Principle of Counting

• If a trial consists of a sequence of choices,
  then the total number of outcomes is found by
  multiplying the number of options in each choice.
• Be careful. Some choices require that you
  consider replacement.

10
Permutations

• A permutation is an ordered arrangement of a
  subset of items.
• Permutations choose without replacement.

11
Combinations

• A combination is an unordered arrangement of a
  subset of items.
• Combinations also choose without replacement.

12
Combinational Analysis

• Combinational Analysis involves combining the
  fundamental counting principle with combinations
  and permutations.

13
Sample Space

• The sample space, S, is the set of all possible
  outcomes.
• S is not a number. S is not necessarily a set of
  numbers.

14
Events

• An event is a subset of the sample space.
• Note that an event is a set, not necessarily a
  single element.
• Two events are mutually exclusive if they cannot
  both happen.
• The intersection of mutually exclusive events is
  the null set.

15
Compliments

• The compliment of an event A is the event that A
  does not occur.
• A and A are mutually exclusive.P(A and A) 0
• P(A or A) 1

16
Venn Diagrams

• Venn Diagrams illustrate events graphically. Each
  event is represented by a circle. The
  intersection of two events is represented by the
  area shared by the corresponding circles

17
Elementary Rules of Probability

• 0 P(A) 1
• P(S) 1
• P(A or B) P(A) P(B) P(A and B)
• If A and B are mutually exclusive, then P(A or B)
  P(A) P(B)
• P(A) P(A) 1 Rule 2?

18
Conditional Probability

• Probability fluctuates with our knowledge of the
  sample space.

19
Multiplication Rule
20
Independence

• Two events are independent if the occurrence of
  one does not effect the occurrence of the other.

21
Try it!

• In a Math 160 class, 50 students received the
  following grades 9 As, 15 Bs, 18 Cs, 5 Ds,
  and 3 Fs. What is the probability that a student
  chosen at random from this class received a
  passing grade (C or better)?
• .84

22
Try it!

• We would like to sit 4 women and 3 men in a row
  so that the women sit in the odd numbered seats.
  How many ways can this be done?
• 144

23
Try it!

• A coin and a die are tossed. Write the following
  events as sets.
• The entire sample space.
• H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6
  
• A three or a four appears on the die.
• H3, H4, T3, T4

24
Try it!

• In a study of regional transportation, officials
  find that the probability is .15 that a worker
  takes the trolley to work, .57 that they drive,
  and .05 that they ride a bike. Find the
  probability that a worker selected at random
• Takes the trolley or drives.
• .72
• Takes the trolley or drives or rides a bike.
• .77
• Does not use one of these means of
  transportation.
• .23

25
Try it!

• Police records for a certain city show that the
  probability a drunk driver will be caught is .45,
  and the probability that if caught, he will be
  convicted is .75. In this city, what is the
  probability that a drunk driver will be caught
  and convicted?
• .3375