Middle Tennessee State University ©2013, MTStatPAL

Empirical probability – based on observed data. Theoretical probability – based on a model of the experiment. Law of large numbers – As the number of repetitions gets large, the empirical probability gets close to the theoretical probability.

The choice of scenario does not affect the underlying probability model.

Probability Rules: Probabilities must be ≥ 0 and ≤ 1. The total probability should equal 1.

Roulette Playing Cards Drop candy on a plate with more than two divisions. Medical outcomes Business decisions Results of scientific measurements

In Roulette, a person wins if the ball falls on the number they have chosen. There are 38 numbers total: 1-36, 0, and 00. Half of 1-36 are red, half are black. 0 and 00 are green.

Assuming the wheel is fair, what is the probability of getting any one specific number? Remembering that there are 38 numbers total, the probability of getting any one of them is 1/38.

What is the probability of not getting a 15? There are 37 numbers that are not 15, and 38 total, so the probability of not getting 15 is 37/38. Notice that P(not 15) = 1 – P(15)

P(E C ) = 1 - P(E) Probability Rules: Probabilities must be ≥ 0 and ≤ 1. The total probability should equal 1.

Suppose you bet on 15. You continue to play, always betting on 15, 100 times. What do you expect to happen?

What is the probability of getting a red number? There are 18 red numbers, and 38 numbers total, so the probability of getting a red number is 18/38.

What is the probability of getting a black number? Is it the same as the probability of not getting a red number?

Suppose you bet on red. You continue to play, always betting on red, 100 times. What do you expect to happen?

One of the bets in Roulette is called a “square bet.” By placing your chip on the square formed by four numbers, you bet on all four of them. One such square contains the numbers 2, 3, 5, and 6.

Suppose you bet on both this square bet (above) and on the red numbers. What is the probability that at least one of your bets pays off?

P(square or red) = P(square) + P(red)

1, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36 3, 5 4, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35 2, 6

P(square or red) = P(square) + P(red) – P(square and red) = 4/ /38 – 2/38 = 20/38

P(E or F) = P(E) + P(F) – P(E and F) If no overlap, this becomes the addition rule for disjoint events: P(E or F) = P(E) + P(F)

Probability Rules: Probabilities must be ≥ 0 and ≤ 1. The total probability should equal 1. Complement Rule: P(E C ) = 1 - P(E) Addition Rules: P(E or F) = P(E) + P(F) – P(E and F) P(E or F) = P(E) + P(F) (If events are disjoint.)