The Nature of Probability.

• Probability helps us understand how likely something is.
• Probability has been formally studied since the 1500s
• It is still an active area of Mathematics
• An experiment is a controlled operation that yields a set of results.
  • Roll a single die
  • Draw three cards from a standard deck of cards.
  • Toss a fair coin
  • Attempt to hit a ball in a baseball game.
  • Roll a ball onto a spinning roulette wheel.
  • Select two marble from a bag of 10 red and 15 blue marbles.
• Outcomes are the possible results of the experiment.
  • the values 1 through 6
  • A-K, hearts, clubs, diamonds, spades, black, red, number, face
  • Heads or Tails
  • Hit or no Hit.
  • 0,00, 1-32, red black green, even odd, ....
  • red, blue
• An event is a subset of the outcomes of an experiment.
  • The results of the roll are less than 3
  • All three cards are face cards
  • The heads show
  • It is a hit.
  • The number 00 shows
  • Two green marbles are selected.
• Empirical probability is determined by performing the experiment and recording the number of times the event occurred.
• Theoretical Probability is determined by examining the outcomes in the event verses the total number of outcomes.
• P(E) = number of times E occurred/ total number of times the experiment was performed.
• 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 relative frequency over the long run that is accurately predictable, not individual events or precise totals.
• Batting average is an empirical measure of the probability of a batter hitting the ball.
  • Batting Average = number of hits/ number of times at bat.
  • Josh Hamilton of the Texas Rangers
  • In the first game of 2010 had a hit for his first three times at bat.
  • His batting average was 3/3 = 1.00
  • Over the season, he had 518 at bats and had 186 hits, his batting average was 186/518 = .359
  • Which of these two do we believe? Why?

Theoretical Probability

• If each outcome of an experiment has the same chance of occurring as any other outcome, then the outcomes are equally likely.
• For an experiment with equally likely outcomes, P(E) = outcomes in E/ total number of outcomes.
• An event E cannot occur if and only if P(E) = 0
• An event E must occur if and only if P(E) = 1
• For every event, 0 ≤ P(E) ≤ 1
• The sum of probabilities of all possible outcomes for an event is 1.
• P(E) + P(not E) = 1
• P(1) = 1-P(not E)
• P(not E) = 1 - P(E)