Chapter 7 Sets

1
Chapter 7 Sets Probability

• Section 7.3
• Introduction to Probability

2
Why use probability?

• A great many problems that come up in the
  applications of mathematics involve random
  phenomena those for which exact prediction is
  impossible.
• The best we can do is determine the probability
  of the possible outcomes.

3
Sample Spaces

• In probability, an experiment is an activity or
  occurrence with an observable result.
• Each repetition of an experiment is called a
  trial.
• The possible results of each trial are called
  outcomes.
• The set of all possible outcomes for an
  experiment is the sample space for that
  experiment.

4

• Example A spinner that is equally divided up
  into three spaces is spun and a coin is tossed.
  Give the sample space for this experiment.
• Spinner Coin Toss Possible Outcomes
• 
  H (1, H)
• 1
• T (1, T)
• H (2, H)
• 2
• T (2, T)
• H (3, H)
• 3
• 
  T (3, T)
• S (1,H), (1,T), (2,H), (2,T), (3,H),
  (3,T)

5
Events

• An event is a subset of a sample space.
• An event in which only one outcome is possible is
  called a simple event.
• If the event equals the sample space, then the
  event is called a certain event.
• If the event is equal to the null, or empty, set,
  then the event is called an impossible event.

6

• Example Consider rolling a single die.
• S 1, 2, 3, 4, 5, 6
• Event A Rolling a 5
• Event B Rolling an odd number
• Event C Rolling a number less than 7
• Event D Rolling a number greater than 6
• Which event is a simple event?
• Event A
• Which event is an impossible event?
• Event D
• Which event is a certain event?
• Event C

7
Set Operations for Events

• Let E and F be events for a sample space, S.
• E ? F occurs when both E and F occur
• E ? F occurs when E or F or both occur
  
• E ' occurs when E does not occur.
• Mutually Exclusive Events
• Events E and F are mutually exclusive events if
  E ? F Ø

8
Probability

• For sample spaces with equally likely outcomes,
  the probability of an event is defined as
  follows.
• Basic Probability Principle
• Let S be a sample space of equally likely
  outcomes , and let event E be a subset of S.
  Then the probability that event E occurs is
• For any event E, 0 P(E) 1.

9

• Example A marble is drawn from a bowl
  containing 3 yellow, 4 white, and 8 blue marbles.
  Find the probability of the following events.
• 1.) A yellow marble is drawn
• P (yellow) 3 / 15 1 / 5
• 2.) A blue marble is drawn
• P (blue) 8 / 15
• 3.) A white marble is not drawn
• P (not white) 11 / 15