Budhi Setiawan, PhD Geostatistik

How to identify the sample space of a probability experiment and to identify simple events How to distinguish between classical probability, empirical probability and subjective probability How identify and use properties of probability Assignment by e-learning

Probability--What is the likelihood of a particular event Some probabilities are easy The mean of rainfall in December will be higher in Palembang than the mean of rainfall in May This is a no brainer but it is possible that we could have a rainy May and dry December that would reverse this

What is the probability that the SFC will win the ISL League this year? What is the probability that the SFC will win the ISL League this year? Sports probability, is very loosely based on statistics Sports probability, is very loosely based on statistics it has too many variables to be scientific it has too many variables to be scientific often it comes down to “I think…” often it comes down to “I think…” We are interested in scientific probability, not how to beat your bookie. We are interested in scientific probability, not how to beat your bookie.

Views of Probability: 1-Subjective: It is an estimate that reflects a person’s opinion, or best guess about whether an outcome will occur. Important in medicine  form the basis of a physician’s opinion (based on information gained in the history and physical examination) about whether a patient has a specific disease. Such estimate can be changed with the results of diagnostic procedures.

2- Objective Classical It is well known that the probability of flipping a fair coin and getting a “tail” is If a coin is flipped 10 times, is there a guarantee, that exactly 5 tails will be observed If the coin is flipped 100 times? With 1000 flips? As the number of flips becomes larger, the proportion of coin flips that result in tails approaches 0.50

2- Objective Relative frequency Assuming that an experiment can be repeated many times and assuming that there are one or more outcomes that can result from each repetition. Then, the probability of a given outcome is the number of times that outcome occurs divided by the total number of repetitions.

Random experiment satisfies following conditions: 1. All possible distinct outcomes are known in advance 2. In any particular experiment outcome is not known in advance 3. Experiment can be repeated under identical conditions The outcome space -  is the set of all possible outcomes. Example 1. Tossing a coin is a random experiment. The outcome space is {H,T} – head and tail. Example 2. Rolling a die. The outcome space is a set - {1,2,3,4,5,6} Example 3. Drawing from an urn with N balls, M of them is red and N-M is white. The outcome space is {R,W} – red and white Example 5. Measuring temperature (in C or in K): What is the outcome space? Something that might or might not happen depending on the outcome of the experiment is called an event. An event is a subset of the outcome space Example: Rolling a die. {1,2,3} or {2,4,6} Example: Measuring temperature in Celsius. Give an example of an event.

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

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

A probability experiment consists of tossing a coin and then rolling a six-sided die. Describe the sample space. There are two possible outcomes when tossing a coin—heads or tails. For each of these there are six possible outcomes when rolling a die: 1, 2, 3, 4, 5, and 6. One way to list outcomes for actions occurring in a sequence is to use a tree diagram. From this, you can see the sample space has 12 outcomes.

{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} | H1T6T5T4T3T2T1H6H5H4H3H H T

A probability experiment consists of recording a response to the survey statement below and tossing a coin. Identify the sample space. Survey: There should be a limit to the number of terms government can serve.  Agree  Disagree  No opinion

A probability experiment consists of recording a response to the survey statement below and tossing a coin. Identify the sample space. A. Start a tree diagram by forming a branch for each possible response to the survey. AgreeDisagree No opinion

A probability experiment consists of recording a response to the survey statement below and tossing a coin. Identify the sample space. B. At the end of each survey response branch, draw a new branch for each possible coin outcome. AgreeDisagree No opinion H H H T T T

A probability experiment consists of recording a response to the survey statement below and tossing a coin. Identify the sample space. C. Find the number of outcomes in the sample space. In this case -- 6 AgreeDisagree No opinion H H H T T T

A probability experiment consists of recording a response to the survey statement below and tossing a coin. Identify the sample space. D. List the sample space {Ah, At, Dh, Dt, Nh, Nt} AgreeDisagree No opinion H H H T T T

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. Even B is rolling at least a 4. B has 3 outcomes: rolling a 4, 5 or 6. Because the event has more than one outcome, it is not simple.

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 and 23, inclusive. A. Decide how many outcomes are in the event. B. State whether the event is simple or not.

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 and 23, inclusive. A. Decide how many outcomes are in the event. The student ’ s age can be {18, 19, 20, 21, 22, or 23} = 6 outcomes B. State whether the event is simple or not. Because there are 6 outcomes, it is not a simple event.

You ask for a student ’ s age at his or her last birthday. Decide whether each even is simple or not: 1. Event D: The student ’ s age is 20. A. Decide how many outcomes are in the event. B. State whether the event is simple or not.

You ask for a student ’ s age at his or her last birthday. Decide whether each even is simple or not: 1. Event D: The student ’ s age is 20. A. Decide how many outcomes are in the event. There is only 1 outcome – the student is 20. B. State whether the event is simple or not. Since there is only one outcome, it is a simple event.

Three types of probability: 1. Classical probability 2. empirical probability 3. subjective probability

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.

P(E) = # of outcomes in E___________ Total # of outcomes in sample space You roll a six-sided die. Find the probability of the following: First when rolling a six-sided die, the sample space consists of six outcomes {1, 2, 3, 4, 5, 6}

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 There is one outcome in event A = {3}. So, P(3) = 1/6 = 0.167

P(E) = # of outcomes in E___________ Total # of outcomes in sample space You roll a six-sided die. Find the probability of the following: 2. Event B: rolling a 7 Because 7 is not in the sample space, there are no outcomes in event B. So, P(7) = 0/6 = 0

P(E) = # of outcomes in E___________ Total # of outcomes in sample space You roll a six-sided die. Find the probability of the following: 3. Event C: rolling a number less than 5. There are four outcomes in event C {1, 2, 3, 4}. So P(number less than 5) = 4/6 = 2/3 ≈0.667

You select a card from a standard 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. A. Identify the total number of outcomes in the sample space. B. Find the number of outcomes in the event. C. Use the classical probability formula.

You select a card from a standard deck. Find the probability of the following: 1. Event D: Selecting a seven of diamonds. A. Identify the total number of outcomes in the sample space. (52) B. Find the number of outcomes in the event. (1) C. Use the classical probability formula. P(52) = 1/52 or

You select a card from a standard deck. Find the probability of the following: 2. Event E: Selecting a diamond A. Identify the total number of outcomes in the sample space. (52) B. Find the number of outcomes in the event. (13) C. Use the classical probability formula. P(52) = 13/52 or 0.25 P(52) = 1/52 or

You select a card from a standard deck. Find the probability of the following: 3. Event F: Selecting a diamond, heart, club or spade. A. Identify the total number of outcomes in the sample space. (52) B. Find the number of outcomes in the event. (52) C. Use the classical probability formula. P(52) = 52/52 or 1.0

Based on observation obtained from probability experiments. The empirical probability of an event E is the relative frequency of event E

A pond containing 3 types of fish: bluegills, redgills, and crappies. Each fish in the pond is equally likely to be caught. You catch 40 fish and record the type. Each time, you release the fish back in to the pond. The following frequency distribution shows your results. Fish Type# of Times Caught, (f) Bluegill13 Redgill17 Crappy10  f = 40 If you catch another fish, what is the probability that it is a bluegill?

The event is “ catching a bluegill. ” In your experiment, the frequency of this event is 13. because the total of the frequencies is 40, the empirical probability of catching a bluegill is: P(bluegill) = 13/40 or 0.325

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. A. Identify the event. Find the frequency of the event. (finding the fraudulent claims, 4) B. Find the total frequency for the experiment. (100) C. Find the relative frequency of the event. P(fraudulent claim) = 4/100 or.04

As you increase the number of times a probability experiment is repeated, the empirical probability (relative frequency) of an event approaches the theoretical probability of the event. This is known as the law of large numbers.

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  f = 1000

P(age 25-34) = 366/1000 = Employee AgesFrequency Age Age Age Age Age and over42  f = 1000

Find the probability that an employee chosen at random is between 15 and 24 years old. P(age 15-24) = 54/1000 =.054 Employee AgesFrequency Age Age Age Age Age and over42  f = 1000

Subjective probability result 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

Classify each statement as an example of classical probability, empirical probability or subjective probability. Explain your reasoning. 1. The probability of your phone ringing during the dinner hour is 0.5 This probability is most likely based on an educated guess. It is an example of subjective probability.

Classify each statement as an example of classical probability, empirical probability or subjective probability. Explain your reasoning. 2. The probability that a voter chosen at random will vote Gerindra is This statement is most likely based on a survey of voters, so it is an example of empirical probability.

Classify each statement as an example of classical probability, empirical probability or subjective probability. Explain your reasoning. 3. The probability of winning a 1000-ticket raffle with one ticket is 1/1000. Because you know the number of outcomes and each is equally likely, this is an example of classical probability.

Classify each statement as an example of classical probability, empirical probability or subjective probability. Explain your reasoning. Based on previous counts, the probability of a salmon successfully passing through a dam on the Columbia River is Event: = salmon successfully passing through a dam on the Columbia River. Experimentation, Empirical probability.

The sum of the probabilities of all outcomes in a sample space is 1 or 100%. An important result of this fact is that if you know the probability of event E, you can find the probability of the complement of event E. The complement of Event E, is the set of all outcomes in a sample space that are not included in event E. The complement of event E is denoted by E ’ and is read as “ E prime. ”

For instance, you roll a die and let E be the event “ the number is at least 5, ” then the complement of E is the event “ the number is less than 5. ” In other words, E = {5, 6} and E ’ = {1, 2, 3, 4} Using the definition of the complement of an event and the fact that the sum of the probabilities of all outcomes is 1, you can determine the following formulas: P(E) + P(E ’ ) = 1 P(E) = 1 – P(E ’ ) P(E ’ ) = 1 – P(E)

Use the frequency distribution given in example 5 to find the probability of randomly choosing an employee who is not between 25 and 34. P(age 25-34) = 366/1000 = So the probability that an employee is not between the ages of is P(age is not 25-34) = 1 – 366/1000 = 634/1000 = 0.634

A. Find the probability that the fish is a redgill. 17/40 =.425 B. Subtract the resulting probability from 1— =.575 C. State the probability as a fraction and a decimal. 23/40 =.575 Fish Type# of Times Caught, (f) Bluegill13 Redgill17 Crappy10  f = 40