Probability Follow-up

Probability Post-Class ActivityWelcome to the post class activity for probability, developed as part of the MTStatPAL project at Middle Tennessee State University.1Review of classEmpirical 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.

If you did the in-class activity for probability, you flipped coins and dropped candies on plates. One goal of the activity was to distinguish between empirical probability, which is based on observed data, and theoretical probability, which is based on a model of the experiment. The law of large numbers tells us that as the number of repetitions of an experiment gets large, the empirical probability gets close to the theoretical probability.2

If you did the in-class activity for probability, you flipped coins and dropped candies on plates. One goal of the activity was to distinguish between empirical probability, which is based on observed data, and theoretical probability, which is based on a model of the experiment. The law of large numbers tells us that as the number of repetitions of an experiment gets large, the empirical probability gets close to the theoretical probability.

3Empirical probabilityHeadsTailsIIIIIIIIII

P(Heads) = 4/10

Empirical probability based on observed data.

4Do we want to mention the formula P(E) = (# of times even E occurs)/(Total # of observed occurances)?Theoretical probability

12Theoretical probability based on a model of the experiment.5Law of LARGE numbers

Review of classThe choice of scenario does not affect the underlying probability model.

You will also have learned that the models for a coin flip and a candy drop are the same. That is, the same probability model can be used for many different random experiments.7Review of classProbability Rules:Probabilities must be 0 and 1.The total probability should equal 1.

Finally, you learned the probability rules, that probabilities are numbers between 0 and 1, and the total probability in the model should be 1.8More than two outcomesRoulettePlaying CardsDrop candy on a plate with more than two divisions.Medical outcomesBusiness decisionsResults of scientific measurements

The experiments that you ran in class had only two outcomes: heads or tails for the coin, and left or right side of the plate for the candy drop. This presentation will focus an experiment with more than two outcomes, roulette. Other experiments with more than two outcomes include playing cards, dropping candy on a plate that is divided into many pieces, and more serious outcomes in medicine, business, and science.9Roulette

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.

The casino game of roulette consists of a large wheel that is spun, and a small ball that is dropped on the wheel. A person wins if the ball falls on the number they have chosen. There are 38 numbers altogether: 1 through 36, 0, and 00. Half of the numbers 1 through 36 are red while the other half are black. 0 and 00 are green.10Roulette

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.Assuming the wheel is fair, what is the probability of getting any one specific number? 11Roulette

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)What is the probability of not getting a 15? Notice that the probability of getting a 15 is 1 out of 38, since it is one specific number. So the probability of not getting a 15 is 1 minus the probability of getting a 15.12Complement RuleP(EC) = 1 - P(E)

Probability Rules:Probabilities must be 0 and 1.The total probability should equal 1.The complement rule says that the probability of getting the opposite of an event is one minus the probability of the event. This rule is now added to the rules we already saw.13Roulette

Suppose you bet on 15. You continue to play, always betting on 15, 100 times. What do you expect to happen?Suppose you bet on 15 and continue to do so 100 times. What do you expect to observe?14Roulette

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.

Suppose instead of betting on just one number, we bet on a group of them. What is the probability of getting a red number?15Roulette

What is the probability of getting a black number? Is it the same as the probability of not getting a red number?What is the probability of getting a black number? Should it be 1-P(red)? Does the complement rule apply here?16Roulette

Suppose you bet on red. You continue to play, always betting on red, 100 times. What do you expect to happen?OK, what if you bet on red 100 times, what do you expect to see?17Roulette

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.

Another way to bet on a group of numbers is to place a square bet. Here, you are betting on four numbers at once, so the probability of winning is 4 out of 38.18Roulette

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?

Suppose you bet on both the square bet and on the red numbers, what is the probability that you win?19Roulette

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

It seems reasonable to decide that the probability of winning at least one of your bets should be the sum of the probabilities of each bet. Is this right? No. 20Roulette

1, 7, 9 , 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 363, 54, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 352, 6The numbers 3 and 5 are in the square bet and are red. So if we add the probabilities, we are adding these in twice.21Roulette

P(square or red) = P(square) + P(red) P(square and red) = 4/38 + 18/38 2/38 = 20/38

It turns out that in order to get the correct probability, we add the two individual probabilities, then subtract out the overlap. For this example, the probability of the square bet winning was 4 out of 38, the probability of the red bet winning was 18 out of 38, and the probability of the overlap was 2 out of 38. So the probability of winning at least one of these bets is 20 out of 38.22General Addition RuleP(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)This gives the general addition rule. To get the probability of at least one event, you add the two probabilities and subtract out the overlap. Notice that if the events do not overlap, we can use the simpler formula here.23RulesProbability Rules:Probabilities must be 0 and 1.The total probability should equal 1.Complement Rule:P(EC) = 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.)

Here is a summary of the rules weve discussed. The probability rules state that probabilities are numbers between 0 and 1 and that the total probability is 1. The complement rule states that the probability of the opposite of an event is one minus the probability of the event. Finally, the addition rules tell us that to get the probability of at least one of two events happening, we add the individual probabilities, and subtract out the overlap, if there is any.24