12.1 The Nature of Probability

• An experiment is a controlled operation that yields a set of results.
  • Roll a fair die and look at the number showing.
  • Select a card at random from a standard deck of cards
  • Flip a coin 5 times.
• The possible results of an experiment are called the outcomes
• • The outcomes of rolling a six sided die are 1-6
  • The outcomes of selecting a card are one of 52 different values
  • The outcomes of tossing a coin are heads or tails.
• An event is a collection of outcomes of an experiment.
  • The result of a die toss is an even number
  • The card selected is a red face card.
  • Five tails occur in a row.
• Empirical probability is calculated by conducting an experiment a number of times.
• Theoretical probability is determined by looking at the set of possible outcomes.
• P(E) = number of times event E has occurred/ Total number of experiments
• The Law of large numbers states that probability statements apply in practice to a large number of trials, not to a single trial. It is the frequency of events in the long run that are predictable, not the occurrence of an individual event.
• Do page 729 problems 15, 17, 25

12.2 Theoretical Probability

• Outcomes are equally likely if all have the same chance of occurring.
• P(E) = number of outcomes favorable to E/ total number of outcomes
• Some Facts:
  • 0 ≤ P(E) ≤ 1 for all events E.
  • If P(E) = 1, the event must occur
  • If P(E) = 0, the event can not occur.
  • The sum of all possible outcomes of an experiment is 1.
• P(A) + P(not A) = 1
• P(not A) = 1-P(A)
• P(A) = 1 - P(not A)
• Do Page 738 problems 13, 14, 17-26, 27,28, 31-34,

12.3 Odds

• Odds have been quoted much longer than the use of formal probability
• There are two types, odds in favor and odds against.
• Odd in favor of E = P(E)/P(not E)
• Odds against E = P(not E) / P(E)
• Given the odds in favor of E, (A:B) P(E) = A/(A+B), P(not E) = B/(A+B)
• On page 745 problem 9, 13-16, 26, 38