Probability

Three Types of Probabilities Theoretical Relative Frequency Personal or Subjective

Determining the Probability of an Outcome Theoretical Probability Example: assume coins made such that they are equally likely to land with heads or tails up when flipped. Probability of a flipped coin showing heads up is ½. Number of desired outcomes/ total number of possible outcomes Relative Frequency Example: observe the number of male and female births in a given city over the course of a year. In 1987 there were a total of 3,809,394 live births in the U.S., of which 1,951,153 were males. Probability of male birth is 1,951,153/3,809,394 = 0.5122

The Relative-Frequency Relative-frequency interpretation: applies to situations that can be repeated over and over again and observed. Examples: Buying a weekly lottery ticket and observing whether it is a winner. Testing individuals in a population and observing whether they carry a gene for a certain disease. Observing births and noting if baby is male or female.

Long-Run Relative Frequency Probability = proportion of time an outcome occurs over the long run Long-run relative frequency of males born in the United States is about 0.512. Possible results for relative frequency of male births: Proportion of male births jumps around at first but starts to settle down just above .512 in the long run.

Tossing a coin n times…. The proportion of heads in “n” tosses of a coin changes as we make more tosses. Eventually it approaches 0.5

Summary of Relative-Frequency Interpretation of Probability Can be applied when situation can be repeated numerous times and outcome observed each time. Relative frequency should settle down to constant value over long run, which is the probability. Does not apply to situations where outcome one time is influenced by or influences outcome the next time. Cannot be used to determine whether outcome will occur on a single occasion but can be used to predict long-term proportion of times the outcome will occur.

Personal Probability Examples: Personal probability: the degree to which a given individual believes the event will happen. Can’t be tested through repetition. The probability must be between 0 and 1 and be coherent. Examples: Probability of finding a parking space downtown on Saturday. Probability that a particular candidate for a position would fit the job best.

Computing theoretical probabilities To calculate the probability of an event, if every outcome is equally likely. Count all the possible outcomes of the random process. Count the outcomes that are favorable to that event The probability is calculated as the ratio # favorable outcomes probability= ----------------------------- # all possible outcomes Example: One deck of cards is shuffled and the top card is placed face down on the table. What is the chance that the card is a king of hearts? How many cards are in a deck? How many king of hearts? Chance=1/52

Computing theoretical probabilities Suppose there are M possible outcomes for one process and N possible outcomes for a second process. The total number of possible outcomes is for the two processes combined is M x N. Examples: How many outcomes are possible when you roll two dice? A restaurant menu offers two choices for an appetizer, five choices for a main course, and three choices for a dessert. How many different three-course meals? A college offers 12 natural science classes, 15 social science classes, 10 English classes, and 8 fine arts classes. How many choices do you have for a schedule comprised of one of each class?

Creating and Using a Sample Space The sample space is the list of all possible outcomes for an event (or events). For example: If recording gender, the sample space is: {male, female} If rolling a die: {1, 2, 3, 4, 5, 6} If rolling a die twice: {(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1)…(6,4) (6,5) (6,6)} --Notice that (1,2) and (2,1) are separate outcomes in the sample space. Order matters!

Creating and Using a Sample Space Let’s take a look at the sample space for a family of three: {BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG} What is the probability of a family with three children having only one boy? {GGB, GBG, BGG}  3/8 What is the probability that the second child in a family with three children will be a boy? {BBB, BBG, GBB, GBG}  4/8 = ½ What is the probability that a family with three children will have only one boy who is also the second child? {GBG}  1/8

Applying Some Simple Probability Rules The sum of the probabilities of all possible outcomes must be one. If A, B, and C are the only possible outcomes: pr(A) + pr(B) + pr(C) = 1 Example: If probability of a single birth resulting in a boy is 0.512, then the probability of it resulting in a girl is 0.488.

Simple Probability Rules If two outcomes cannot happen simultaneously, they are said to be mutually exclusive. The probability of one or the other of two mutually exclusive outcomes happening is the sum of their individual probabilities. pr(A or B) = pr(A) + pr(B) Example: If you think probability of getting an A in your statistics class is 50% and probability of getting a B is 30%, then probability of getting either an A or a B is 80%. Thus, probability of getting C or less is 20%.

Simple Probability Rules If two outcomes could happen simultaneously, they are not mutually exclusive. The probability of one or the other of two mutually exclusive outcomes happening is the sum of their individual probabilities minus the probability that they both happen. pr(A or B) = pr(A) + pr(B) – pr(A and B) Example: What is the probability of getting a number that is even or a multiple of 3 on one roll of a die? The probability of rolling an even number is 3/6. The probability of rolling a multiple of 3 is 2/6. But the probability of rolling a number that is both even and a multiple of 3 is 1/6. So the probability of rolling either a even number or a multiple of 3 is 3/6 + 2/6 – 1/6 = 4/6

Simple Probability Rules If two events do not influence each other, the events are said to be independent of each other. If two events are independent, the probability that they both happen is found by multiplying their individual probabilities. pr(A and B) = pr(A)×pr(B) Example: A woman will have two children. Assume the probability the birth results in boy is 0.512. Then probability of having 2 boys in a row is 0.512×0.512 = 0.262144. About a 26% chance a woman will have 2 boys.

Simple Probability Rules The probability of getting at least one of a possible outcome out of numerous trials is equal to 1 minus the probability of not getting that outcome at all. pr(at least one) = 1 – pr(none) Example: If a die is rolled twice, what is the probability that at least one of the rolls is a 2? The probability of not rolling a 2 both times = 5/6 × 5/6 = 23/36 1- 25/36 = 11/36

Simple Probability Rules Example: You purchase 10 lottery tickets, for which the probability of winning some prize is 1 in 10. What is the probability that you will have at least one winning ticket? Probability of winning something is 1/10 = 0.1 Probability of winning nothing is 9/10 = 0.9 P( at least one winner in 10 tickets) = 1 – P( losing 10 times in a row) = 1 – (.9)*(.9)*(.9)*(.9)*(.9)*(.9)*(.9)*(.9)*(.9)*(.9) 1 - 0.910 = 0.651

Simple Probability Rules Another Example: What is the probability that a region will experience at least one 100-year flood during the next 100 years? Probability of a flood is 1/100. Probability of no flood is 99/100. P( at least one flood in 100 years) = 1 - .99100 = 0.634

Answer the following questions Answer the following questions. Determine whether each is asking for a theoretical, relative frequency, or personal probability? If you flip a coin and do it fairly, what is the probability that it will land heads up? What is the probability that you will eventually own a home; that is, how likely do you think it is? A bag contains some marbles. You pull one marble 10 times and get a red marble 6 times. What is the probability of getting a red marble?

Explain what’s wrong with this partial answer to one of the previous questions: “The probability that I will eventually own a home is 1/2 because either it will happen or it won’t.”