WARM – UP A two sample t-test analyzing if there was a significant difference between the cholesterol level of men on a NEW medication vs. the traditional yields a p-value of 0. DEFINE p-value in context a problem. P-Value = the Probability that you would obtain or even more extreme results given the assumption H0 is True. “There is a zero chance of seeing a difference between the two medications given the two drugs are actually equally effective.”

Warm – Up (continued…) 2. What does a Significance level of 0.05 mean? α = 0.05 means you would only obtain these results 5 out of 100 times or 1 time out of every 20 samples. RARE!!

Chapter 14 - PROBABILITY We call a phenomenon RANDOM if individual outcomes are uncertain. The PROBABILITY of any outcome is the proportion of times the outcome would occur in a very long series of Independent repetitions or trials. Relative Frequency. An EVENT = P(A)

The Law of Large Numbers says that the long-run relative frequency of repeated independent events settles down to the true probability as the number of trials increase. If you flip a fair coin 19 times and it come up tails. What is the probability that it will be tails on the 20th flip? Many people believe in The Law of Averages in which they feel that in the short term random phenomena is supposed to compensate somehow for whatever happened in the past.

PROBABILITY MODELS The SAMPLE SPACE, S, of a random phenomenon is the set of all possible outcomes. EXAMPLE: List the sample space for the event of tossing a coin and roll one die. S = { H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6 }

MULTIPLICATION PRINCIPLE If one event can be done ‘a ‘ ways and another event ‘b’ ways, then the total number of outcomes in the sample space = a x b . EXAMPLE: What is the probability of tossing a coin and getting “heads” 5 times in a roll? EXAMPLE: A 6 question multiple choice test has four answer choices for each question. If you are completely unprepared, what is the probability that you will get all 6 questions correct?

P(1 and 1 and 1) = P(Not 1 and Not 1 and Not 1) = What is the Probability of rolling a die and getting a “1” three times in a row? P(1 and 1 and 1) = What is the Probability of rolling a die and NOT getting a “1” at all in three rolls? P(Not 1 and Not 1 and Not 1) =

Chapter 14 - More Probability PROBABILITY RULES: The Probability of P(A) is always: 0 ≤ P(A) ≤ 1 . The Probability of every event in the Sample Space together is equal to one. P(S) = 1 . The Complement of any event is the Probability that the event will NOT occur. P(Ac) = 1 – P(A) . Two events A and B are Disjoint if they have no outcomes in common and so can never occur simultaneously. The Addition Rule for disjoint events is: P(A or B) = P(A) + P(B).

Sample spaces can also be modeled by TREE DIAGRAMS. EXAMPLE: List the sample space for the event of tossing a coin and roll one die. 1 2 3 4 H 5 6 1 2 3 4 T 5 6 What is the Probability of Tossing a coin, getting “TAILS” and then rolling a die and getting an “ODD” number? P(TAIL and ODD#) =

Textbook: Page 339: 1-6, 8

Textbook: Page 339: 1-6, 8 Roulette. If a roulette wheel is to be considered truly random, then each outcome is equally likely to occur, and knowing one outcome will not affect the probability of the next. Additionally, there is an implication that the outcome is not determined through the use of an electronic random number generator. 2. Rain. When a weather forecaster makes a prediction such as a 25% chance of rain, this means that when weather conditions are like they are now, rain happens 25% of the time in the long run. 3. Winter. Although acknowledging that there is no law of averages, Knox attempts to use the law of averages to predict the severity of the winter. Some winters are harsh and some are mild over the long run, and knowledge of this can help us to develop a long-term probability of having a harsh winter. However, probability does not compensate for odd occurrences in the short term. Suppose that the probability of having a harsh winter is 30%. Even if there are several mild winters in a row, the probability of having a harsh winter is still 30%.

Textbook: Page 339: 1-6, 8 4. Snow. The radio announcer is referring to the “law of averages”, which is not true. Probability does not compensate for deviations from the expected outcome in the recent past. The weather is not more likely to be bad later on in the winter because of a few sunny days in autumn. The weather makes no conscious effort to even things out, which is what the announcer’s statement implies. 5. Cold streak. There is no such thing as being “due for a hit”. This statement is based on the so-called law of averages, which is a mistaken belief that probability will compensate in the short term for odd occurrences in the past. The batter’s chance for a hit does not change based on recent successes or failures.