16.1: Basic Probability

Definitions Probability experiment: An action through which specific results (counts, measurements, or responses) are obtained. Outcome: The result of a single trial in a probability experiment. Sample Space: The set of all possible outcomes of a probability experiment Event: One or more outcomes and is a subset of the sample space

How it works: Probability experiment: Roll a six-sided die Sample space: {1, 2, 3, 4, 5, 6} Event: Roll an even number {2, 4, 6} Outcome: Roll a 2, {2}

Definitions in Play ExperimentExample of EventComplete Sample Space Single Birth1 Female{f, m} 3 Births2 Females and 1 Male{fff, ffm, fmf, fmm, mff, mfm, mmf, mmm} Simple Event Not a Simple Event

Example Construct the sample space of an experiment that involves flipping a coin and then rolling a die.

Simple events Simple Event: An event that consists of a single outcome. Decide whether the event is simple or not. Explain your reasoning: 1.For quality control, you randomly select a computer chip from a batch that has been manufactured that day. Event A is selecting a specific defective chip. Simple because it has only one outcome: choosing a specific defective chip. So, the event is a simple event. 2. You roll a six-sided die. Event B is rolling at least a 4. Because the event has more than one outcome, it is not simple.

More Examples You ask for a student’s age at his or her last birthday. Decide whether each even is simple or not: 1. Event C: The student’s age is between 18 & Event D: The student’s age is 20.

Probability Type 1: Classical P(E) = # of outcomes in E___________ Total # of outcomes in sample space You roll a six-sided die. Find the probability of the following: 1. Event A: rolling a 3 2. Event B: rolling a 7 3. Event C: rolling a number less than 5.

More Examples You select a card from a standard 52-card deck. Find the probability of the following: 1. Event D: Selecting a seven of diamonds. 2. Event E: Selecting a diamond 3. Event F: Selecting a diamond, heart, club or spade.

Probability Type 2: Empirical Relative Frequency Approximation of Probability: Conduct or observe a procedure, and count the number of times that event A actually occurs. Based on these actual result, P(A) is approximated as follows: P(A) = Number of times A occurred Number of times the procedure was repeated

Practice For a recent year, there were 6,511,100 cars that crashed among the 135,670,000 cars registered in the U.S. Find the probability that a randomly selected car in the U.S will be in a crash this year.

Practice An insurance company determines that in every 100 claims, 4 are fraudulent. What is the probability that the next claim the company processes is fraudulent.

Using frequency distributions to find Probability You survey a sample of 1000 employees at a company and record the ages of each. The results are shown below. If you randomly select another employee, what is the probability that the employee is between 25 and 34 years old? Employee AgesFrequency Age Age Age Age Age and over42

Probability Type 3: Subjective Subjective probability results from intuition, educated guesses, and estimates. For instance, given a patient’s health and extent of injuries, a doctor may feel a patient has 90% chance of a full recovery. A business analyst may predict that the chance of the employees of a certain company going on strike is.25 A probability cannot be negative or greater than 1, So, the probability of and event E is between 0 and 1, inclusive. That is 0  P(E)  1

Complementary Events The complement of event A, denoted by consists of all outcomes in which event A does not occur.

Practice A typical question on the SAT requires the test taker to select one of five possible choices: A, B, C, D, or E. Because only one answer is correct, if you make a random guess, your probability of being correct is 1/5. Find the probability of making a random guess and being incorrect.

Define the Complement Event A: Getting at least 1 strike when bowling. Event B: Not being in the age group. Event C: Rolling an even number on a die.

Team Practice

Trashketball Rules Students will be split into teams of 4 Each student will be assigned a letter A-D Students will work on the problems together, and one random student will be called on to present an answer One person from one group will randomly be selected to explain the problem to the class If the student is correct, they will get a tally point (which is equivalent to one shot) Scoring rounds will come after every three questions

Round 1 When studying the affect of heredity on height, we can express each individual genotype, AA, Aa, aA, aa, on an index card and shuffle the four cards and randomly select one of them. What is the probability that we select a genotype in which the two components are different?

Round 2 The internet service provided AOL asked users this question about Kentucky Fried Chicken (KFC): “Will KFC gain or lose business after eliminating trans fats?” Among the responses received, 1941 said that KFC would gain business, 1260 said that KFC business would remain the same, and 204 said that KFC would lose business. Find the probability that a randomly selected response states that KFC would gain business.

Round 3 If a year is selected at random, find the probability that Thanksgiving Day will be (a) on a Wednesday or (b) on a Thursday.

Shots behind the closer line to the trash can are worth 2 points Shots behind the closer line to the trash can are worth 2 points Shots behind the further line to the trash can are worth 3 points Shots behind the further line to the trash can are worth 3 points

Round 4 Find the probability that when a couple has 3 children, they will have exactly 2 boys. Assume that boys and girls are equally likely and that the gender of any child is not influenced by the gender of any other child.

Round 5 The table below summarizes the test results for 98 different subjects. In each case, it was known whether or not the subject lied. Assume that one of the 98 test results is randomly selected, find the probability that it is a positive result. Did the Subject Actually Lie? No (Did Not Lie)(Yes) Lied Positive Test Result (Polygraph test indicated that the subject lied) 15 (false positive) 42 (true positive) Negative Test Result (Polygraph test indicated that the subject did not lie). 32 (true negative) 9 (false negative)

Round 6 When predicting the chance that we will elect a Republican President in the year 2016, we could reason that there are two possible outcomes (Republican, not Republican), so the probability of a Republican president is ½ or 0.5. Is this reasoning correct? Why or why not?

Shots behind the closer line to the trash can are worth 2 points Shots behind the closer line to the trash can are worth 2 points Shots behind the further line to the trash can are worth 3 points Shots behind the further line to the trash can are worth 3 points