Probability Basic Concepts Start with the Monty Hall puzzle We deal with probability almost every day. You frequently make decisions based on the most likely outcome from multiple possibilities. Weather is a good example. In business, if a company has been losing money, consistently, over the past 5 years, would it be a good investment? If you were playing a game and tossing a coin and the coin came up heads 4 times in a row, would you assume the coin was not fair? We’re going to work on the probability rules that will help you understand the above situations.

Experiment An activity or measurement that results in an outcome we cannot predict with absolute certainty. A coin is flipped twice. We conduct a 10 year medical study on 12,000 participants The definition of an experiment in statistics is broader than the one used in science. A scientific experiment involves scientific instrumentations such as thermometers, microscopes, telescopes and tubes. A statistical experiment may involve all these items but it mainly involves recording data and measurements.

Event The simplest outcome of an experiment Toss two coins: 4 possible outcomes for each toss Each possible outcome is an event All the possible outcomes comprise the… A simple event is the simplest outcome of an experiment.

Sample space All the possible outcomes of an experiment: More examples (HH) (HT) (TH) (TT) More examples One coin One die Red, white, and blue socks With and without replacement A collection of all the simple events. Typically denoted by S. We want to investigate the sample space of rolling a die and the sample space of tossing a coin. Solution: As we know, there are 6 possible outcomes for throwing a die. We may get 1, 2, 3, 4, 5, or 6. So we write the sample space as the set of all possible outcomes: S = {1, 2, 3, 4, 5, 6} Similarly, the sample space of tossing a coin is either head (H) or tail (T) so we write S = {h,t} With socks: Diagram on board of with and w/out, then create sample spaces

Probability measure A function, P, from the subsets of the sample space (Ω) that satisfies the following axioms: P(Ω) = 1; that is, the sum of all the probabilities in the sample space will = 1 P(A)≥0 for any event, A, that is included in the sample space (Ω). For mutually exclusive events A1 and A2 within the same sample space P(A1UA2) = P(A1) + P(A2) Probability - A number between 0 and 1 that expresses the chance that the event will occur. P= 0 or P = 1 are not likely (no reason to compute them) Read the last one as or! Examples: Experiment: Tomorrow’s trading on the NYSE Sample space: DJIA goes up 30 or more points Event = A = the Dow goes up 30 points Probability: The chance (0-1) of A happening Experiment: Roll one die Sample space: 1-6 Event = A = Less than a 3 Probability: P(A) = 1/3

Classical Probability Proportion of times an event can be expected to occur All outcomes are equally likely: Describes probability in terms of proportion of times event can be expected to occur. Works when all outcomes are equally likely. P=(# of possible outcomes when the event occurs/# of possible outcomes)

Examples Flip a coin twice: Roll a die: P(HH) = P(HT) = P(TH) = P(TT) = 0.25 Roll a die: P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6 Examples: Dice Example: 2 coins, P(HH)=1/4 or .25 Dice, P(2,3,or4) =3/6 or .5

Examples Consider events A and B (coins) We can determine their probabilities by adding the probability of each occurrence. A: exactly one head [(HT), (TH)] P(A) = 0.25 + 0.25 = 0.50 B: first flip head [(HH),(HT)] P(B) = ¼ + ¼ = ½ If we have an experiment where we roll one die, what is the probability of rolling less than a 3? Sample space Event P(A)

Compound Events Mutually Exclusive Events Exhaustive Events If one event occurs, the other cannot Exhaustive Events Set of events that include all the possible outcomes of an experiment Events are exhaustive because one of them must occur When events are mutually exclusive and exhaustive, sum of their probabilities must equal 1 Examples of mutually exclusive: Boy or girl, heads or tails, Ace or king, etc.

Intersection and Union Intersection of Events Two or more events occur at the same time in the same experiment. [A and B, or A and B and C] Union of Events At least one of a number of possible events occurs in the same experiment. [A or B, or A or B or C]

What’s the probability? Experiment: the throw of one die A: (observe an even number) B: (observe a number <=3) Describe A union B Describe A intersection B Calculate the probabilities of 1. and 2. Solution: The sample space of a fair die is S = {1, 2, 3, 4, 5, 6}. The sample spaces of the events A and B above are S(A) = A = {2, 4, 6} and S(B) = B = {1, 2, 3} . 1. The union of A and B is the event if we observe either an even number, a number that is equal to 3 or less, or both on a single toss of the die. In other words, the simple events of A ∪ B are those for which A occurs, B occurs or both occur: A ∪ B = {2, 4, 6} ∪ {1, 2, 3} = {1, 2, 3, 4, 6} 2. The intersection of A and B is the event that occurs if we observe both an even number and a number that is equal to or less than 3 on a single toss of a die. A ∩ B = {2, 4, 6} ∩ {1, 2, 3} = {2} In other words, the intersection of A and B is the simple event to observe a 2. 3. Remember the probability of an event is the sum of the probabilities of the simple events, P(A ∪ B) = P(1) + P(2) + P(3) + P(4) + P(6) = 1 6

Example: Unions and Intersections Age Under 15 (B) 15 or older (B’) Male (A) 3477 5436 8913 Female (A’) 1249 1287 2536 4726 6723 11449 A survey of the victims of accidents involving fireworks. First, convert these to relative frequencies Then draw ven diagrams showing the following: Mutually exclusive events: male = A female = A’, B or B’ Exhaustive events. The 4 mutually exclusive events are also exhaustive because a victim must be in one of the 4 groups Are the 4 ME events exhaustive? Intersction: Ven – show intersection of A (males = 8913) and B (under 15 = 4726) = 3477. Complement = 1287 (females over 15) Union: Ven – show A or B (male or under 15 = 10162) Complement female over 15.

Addition Rules When events are mutually exclusive When events are not mutually exclusive Both of these formulas say the same thing. When things are mutually exclusive A intersection B = 0 Mutually exclusive – go back one slide What is the probability that a household is in the south or the west? Not mutually exclusive. Go back to slide 10. P(male) = .475+.304=.779 P(under 15) = .109+.304 = .413 Don’t work because we counted the intersection twice. We need to subtract it out, once. (-.304)

Practice Access to Internet? Region Northeast Midwest South West Yes 9.7 11.8 16.9 12.2 50.6 No 1.2 2.3 3.8 2.1 9.4 10.9 14.1 20.7 14.3 60.0 Identify What is the probability that a household would be in the South or Midwest or have internet access? What is the probability a household would be in the West and not have Internet access? Identity 2 events that are mutually exclusive Identify 2 events that intersect

Practice Access to Internet? Region Northeast Midwest South West Yes 9.7 11.8 16.9 12.2 50.6 No 1.2 2.3 3.8 2.1 9.4 10.9 14.1 20.7 14.3 60.0 What is the probability that a household would be in the South or Midwest or have internet access? What is the probability a household would be in the West and not have Internet access?

Venn Diagram What is the probability of Ac? 0.05 B = 0.15 What is the probability of Ac? What is the probability of Bc? What is the probability of AUB?

Probabilities Marginal Probability Joint Probability Probability a given event will occur. No other events are considered. P(A) Joint Probability Probability that two or more events will all occur. P(A and B) Conditional Probability Probability that an event will occur given that another event has already occurred. P(A|B) We use addition rules to calculate the probability that at least one of several events will occur, and multiplication rules to determine the probability that 2 or more events will all occur.

Multiplication Rules Independent Events Dependent Events The occurrence of one has no effect on the probability that the other will occur. Dependent Events The occurrence of one event influences the probability of the other. Independent: flip twice, heads and tails Dependent: We’ll go back to the fireworks chart. Being male increases the probability of injury.

Multiplication Rules When events are independent When events are not independent Show a tree diagram Coin tosses (x3) Free throws. 1: 70% 2: 70%, but if he makes first, 80%, if he misses, 60%

Fireworks Chart Age Under 15 (B) 15 or older (B’) Male (A) 3477 .304 5436 .475 8913 .779 Female (A’) 1249 .109 1287 .112 2536 .221 4726 .413 6723 .587 11449 1.000

Conditional Probability Type of Policy (%) Category Fire Auto Other Total % Fraudulent 6 1 3 10 Legitimate 14 29 47 90 Total 20 30 50 100 Examination of a large number of insurance claims. You are responsible for checking fraudulent claims. What is the probability that the next claim you will examine is fraudulent? P(F) = .10 Additional information – type of policy. Suppose you know that the next claim you examine is from a fire policy. What’s the probability of F given Fire Policy? .30

Conditional Probability Problem A corporation is going to select 2 of its regional managers for promotion to VP. They have 6 male and 4 female regional managers. Assume each manager has an equal probability (1/10) of being selected. What is the probability that both people selected for regional manager are male? A corporation is going to select 2 of its regional managers to the office of VP. They have 6 male and 4 female RMs. Assume all are equally qualified so the probability of any one being promoted is 0.1. What is the probability that both selected are male? P(A) = 1st selection is male = .6 P(B) = 2nd selection is male P(B|A) = 5/9 P(A int B) = P(A)P(B|A) = 6/10x5/9 = 1/3

Practice Problems A fair coin is tossed 4 times. What is the probability of getting at least one tail? What is the probability of getting exactly one head? A card is drawn for a standard deck. What is the probability that card will be a jack or a king? This is a good time to discuss hold ‘em probabilities.

More Practice Problems A standard pair of 6-sided dice is rolled. What is the probability of rolling a sum greater than or equal to 3? Three cards are drawn with replacement from a standard deck. What is the probability that the 1st card will be a diamond, the 2nd card will be black, and the third card will be a queen?

Still more practice problems 2 cards are drawn without replacement from a standard deck. What is the probability of choosing a club and then a black card? A box contains 6 green marbles and 19 white marbles. If the first marble chosen is white, what is the probability of choosing a white marble on the second draw? IMPORTANT: In the homework problem (see #30) this is WITHOUT REPLACEMENT

Counting Principle of multiplication m ways for event 1 to happen n ways for event 2 to happen Total number of possibilities = m x n If each of k independent events can happen n different ways, the total number of possibilities is nk INTRODUCE COUNTING PROBLEMS WITH THE BIRTHDAY PARADOX If there are n ways in which event one can happen and m ways for event two to happen, the total number of possibilities is m x n. For example, if you have 2 cars and 4 different routes you can drive to school, there are 8 possible ways to get here. Or, assume there are 3 options, cars with 6 colors, 2 transmissions, 3 engine types. Example for part 2 – auto license w/ 6 different numbers or letters.

Counting Factorial rule of counting n! = n x (n-1) x (n-2) x … x 1 0! = 1 Example: lining up all the people in the classroom Put the things in buckets.

Counting Permutations: Number of possible arrangements of n items in order In a permutation, each item can appear only once, and each order of the item’s arrangement constitutes a separate permutation. Let’s say I have 6 books and can only put 4 on a shelf, how many ways can I arrange these books? (360) Remember, each different arrangement is a different permutation. [ABCD and DCBA are different] I have 8 different tests to grade, but only have time to grade 5 of them today. In how many different orders can I grade the 5 tests?

Counting Combinations: Order doesn’t matter. We consider only the possible set of objects. Re-do the test example, but don’t consider the order in which the tests are graded. [ABCD is NOT different from BDCA]

Counting Problems A 29-sided die is rolled 2 times. How many different outcomes are possible? License plates consist of 2 letters followed by three numbers. Duplicate digits are allowed. How many different outcomes are possible? A doctor visits her patients during morning rounds. In how many ways can she visit the 8 patients? Start with the examples of the game: 3 letter words 4 letter words What is the probability my 6 letter word will be spelled properly

More Counting Problems A coordinator will select 6 songs from a list of 8 songs to compose a musical lineup. How many different lineups are possible? 4 cards are chosen from a standard deck. How many different 4 card hands are possible? A person tosses a coin 16 times. In how many ways can he get 6 heads?

Practice The daily number in a state lottery is a 3-digit integer between 000 and 999. What is the probability that the winning number will be 555? Today’s winning number is 347. You are going to buy a ticket tomorrow and you plan to select number 347. Is this a good idea? Why or why not?

Practice Neither computer system will be operational. Your company has two computer systems available for processing telephone orders. Computer system A has a 10% chance of being down; computer system B has a 5% chance of being down. The computer systems operate independently. At least one system needs to work in order to process new orders. For a typical telephone order, determine the probability that: Neither computer system will be operational. Both computer systems will be operational. Exactly one of the computer systems will be operational. What is the probability that the order can be processed without delay?

Practice A security service employing 10 officers has been asked to provide 3 officers for crowd control at a local carnival. In how many different ways can the firm staff this event? If you need one more, try the Word game