Chapter 6: Probability and Simulation The study of randomness
6.1 Randomness Probability describes the pattern of chance outcomes. Probability is the basis of inference Meaning, the pattern of chance outcomes influences the predictions made and conclusions drawn in a scenario.
Handout: #1 From one trial to another, we cannot predict the probability of an event. However, when we perform more and more trials, the probability stabilizes and changes little.
Handout #2: Haphazard? While random is often a synonym for haphazard in conversation, in statistics with more and more repetitions, a random phenomenon approaches a long-term regularity. We usually don t have the opportunity to see enough repetitions to see the probabilities stabilize. Instead, with seeing the event happen once or twice, we interpret the occurrence as random, which makes it seem haphazard.
Handout #3 (from pg 314) The idea of probability is empirical, which means based on observation.
Handout: #4 In attempt to estimate the empirical probability of tossing a coin and counting the proportion of heads, Count Buffon (from the 18 th century) tossed a coin 4040 times, Karl Pearson tossed a coin 24,000 times around 1900, and John Kerrich tossed a coin 10,000 times while imprisoned by the Germans during WWII. Karl Pearson tossed a coin the most, but each pioneer of coin-tossing got heads about half the time (Buffon: 0.5069, Pearson: 0.5005, Kerrich: 0.5067)
Handout #5: Randomness The outcome of one trial must not influence the outcome of an other the trials must be independent. Probability is empirical (observed by many trials) Simulations lead us to probability; tools like number generators can help make long runs of trials.
Handout #6 Probability is used to describe life span, measurements, traffic flow, genetic makeup, spread of epidemics, setting insurance rates, election predictions where does it stop?!
6.2 Probability Models Note that probability models have two parts: A list of possible outcomes A probability for each outcome.
Sample Space A sample space is the set of all possible outcomes. To specify S we must state what constitutes an individual outcome, then which outcomes can occur (can be simple or complex) Simple ex: coin tossing, S = {H, T} Complex ex: US Census: If we draw a random sample of 50,000 US households, as the survey does, the S = (all 50,000 households}
Sample Space: Rolling two dice At a casino- 36 possible outcomes when we roll 2 dice and record the up-faces in order (first die, second die) Gamblers care only about number of dots face up so the sample space for that is: S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Sample Space: A deck of cards S = {you would list out all 52}
Techniques for finding outcomes Tree diagram: lists outcomes in an organized way For tossing a coin then rolling a die Starting from a point, each branch represents the possible outcomes from the occurrence of the given event. Be sure to list the set of outcomes from the series of events here: S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
TREE DIAGRAM example a. Create a tree diagram for flipping 2 coins. List the sample space as S = { } a. Create a tree diagram for flipping 4 coins. List the sample space as S = { }
Multiplication Principle used if you want to calculate THE NUMBER of outcomes it doesn t tell you what the outcomes are Ex: There are 12 outcomes when flipping a coin and then rolling a die: 2 possible outcomes when a coin is flipped, 6 possible outcomes when rolling a die so 2 x 6 = 12
EXAMPLE of multiplication principle Confirm that there are 16 outcomes when flipping 4 coins. Your city has grown and has added a new phone number. In addition to phone numbers that start 434-, now your town has phone numbers that start as 545. How many phone numbers are added to your town?
With/without replacement Whether or not you replace an object back into the population to sample from again, affects the number of outcomes in your sample space. Ex: If you take a card from a deck of 52, don t put it back, then draw your 2 nd card etc., that s without replacement. If you take a card, write it down, put it back, draw 2 nd card etc., that s with replacement.
EXAMPLE of replacement Your city has grown and has added a new phone number. In addition to phone numbers that start 434-, now your town has phone numbers that start as 545. How many phone numbers are added to your town if you can t have repeated digits in the last four?
Event An event is an outcome or set of outcomes of a random phenomenon. It is a subset of the sample space. Ex: When flipping four coins, exactly two heads is an event. So looking at the earlier example, what outcomes constitute this event? We ll call the event A. A = {HHTT, HTHT, HTTH, THHT, THTH, TTHH}
Probability Rules In words and notation 1. Any probability is a number between 0 and 1 P(A) of any event A satisfies 0 P(A) 1 2. The sum of the probabilities of all possible outcomes = 1 If S = sample space in a probability model, then P(S) = 1 3. The probability that an event doesn t occur (called the complement of event A) is 1 minus the probability that it does occur P(A^c) = 1 P(A) 4. If 2 events have no outcomes in common (called disjoint, they can t occur together), the probability that one OR the other occurs is the sum of their individual probabilities P(A or B) = P(A) + P(B) (This is called the addition rule for disjointed events)
Venn diagrams help visualize the relationship between events Disjoint events A and B Complement A^c of an event A
Example of disjoint/complement Select a woman aged 25 29 years old at random and record her marital status. At random means that we give every such woman the same chace to be the one we choose. We choose an SRS of size 1. The probability of any marital status is just the proportion of all women aged 25 to 29 who have that status; if we selected many women, this is the proportion we would get. Here is the probability model Marital Status Never married Married Widowed Divorced Probability 0.353 0.574 0.002 0.071 Find the probability that the woman we draw is not married, using The complement rule The addition rule
Probabilities in a finite space Looking at the probability model re: marital status of women, notice the sum of the separate events. The probabilities for the events ended up being unique numbers, but if two events have the same probability, they are labeled as equally likely.
When to add, when to multiply The addition rule for disjointed events is used when finding the probability of one event occurring. Event A OR B. If finding the probability that two events occur, the probabilities of these events are multiplied.
Independence & the Multiplication Rule To find the probability for BOTH events A and B occurring The multiplication rule applies only to independent events; can t use it if events are not independent! In a Venn diagram, the event {A and B}is represented in the overlap
Independent or not? Examples Coin toss I: Coin has no memory and coin tossers cannot influence fall of coin Drawing from deck of cards NI: First pick, probability of red is 26/52 or.5. Once we see the first card is red, the probability of a red card in the 2 nd pick is now 25/51 =.49 Taking an IQ test twice in succession NI
Multiplication Rule Example 1 A general can plan a campaign to fight one major battle or three small battles. He believes that he has probability 0.6 of winning the large battle and probability 0.8 of winning each of the small battles. Victories or defeats in the small battles are independent. The general must win either the large battle or all three small battles to win the campaign. Which strategy should he choose?
Multiplication Rule Example 2 A diagnostic test for the presence of the AIDS virus has the probability of 0.005 of producing a false positive. That is, when a person free of the AIDS virus is tested, the test has probability 0.005 of falsely indicating that the virus is present. If all 140 employees of a medial clinic are tested and all 140 are free of AIDS, what is the probability that at least one false positive will occur?
More applications of Probability Rules If two events A and B are independent, then their complements are also independent. Ex: 75% of voters in a district are Republicans. If an interviewer chooses 2 voters at random, the probability that the first is a Republican and the 2 nd is not a republican is.75 x.25 =.1875
6.3 General Probability Rules
Addition Rule for Disjoint events
General Addition rule for Unions of 2 events
Example: Deb and Matt are waiting anxiously to hear if they ve been promoted. Deb guesses her probability of getting promoted is.7 and Matt s is.5, and both of them being promoted is.3. The probability that at least one is promoted =.7 +.5 -.3 which is.9. The probability neither is promoted is.1. The simultaneous occurrence of 2 events (called a joint event, such as deb and matt getting promoted) is called a joint probability.
Conditional Probability The probability that we assign to an event can change if we know some other event has occurred. P(A B): Probability that event A will happen under the condition that event B has occurred. Ex: Probability of drawing an ace is 4/52 or 1/13. If your are dealt 4 cards and one of them is an ace, probability of getting an ace on the 5 th card dealt is 3/48 or 1/16 (conditional probability- getting an Ace given that one was dealt in the first 4).
In words, this says that for both of 2 events to occur, first one must occur, and then, given that the first event has occurred, the second must occur.
Remember: B is the event whose probability we are computing and A represents the info we are given.
Extended Multiplication rules The union of a collection of events is the event that ANY of them occur The Intersection of any collection of events is the event that ALL of them occur
Example Only 5% of male high school basketball, baseball, and football players go on to play at the college level. Of these only 1.7% enter major league professional sports. About 40% of the athletes who compete in college and then reach the pros have a career of more than 3 years. Define these events: A = competes in college B = competes pro C = pro career longer than 3 years P(A) =.05 P(B A) =.017 P(C A and B) =.400 What is the probability a HS athlete will have a pro career more than 3 years? The probability we want is therefore P(A and B and C) = P(A)P(B A)P(C A and B) =.05 x.017 x.40 =.00034 So, only 3 of every 10,000 high school athletes can expect to compete in college and have a pro career of more than 3 years.
Extended tree diagram + chat room example 47% of 18 to 29 age chat online, 21% of 30 to 49 and 7% of 50+ Also, need to know that 29% of all internet users are 18-29 (event A1), 47% are 30 to 49 (A2) and the remaining 24% are 50 and over (A3). What is the probability that a randomly chosen user of the internet participates in chat rooms (event C)? Tree diagram- probability written on each segment is the conditional probability of an internet user following that segment, given that he or she has reached the node from which it branches. (final outcome is adding all the chatting probabilities which =.2518)
Bayes Rule Another question we might ask- what percent of adult chat room participants are age 18 to 29? P(A1 C) = P(A1 and C) / P(C) =.1363/.2518 =.5413 *since 29% of internet users are 18-29, knowing that someone chats increases the probability that they are young! Formula sans tree diagram: P(C) = P(A1)P(C A1) + P(A2)P(C A2) + P(A3)P(C A3)
6.3 Need to Know summary(print) Complement of an event A contains all outcomes not in A Union (A U B) of events A and B = all outcomes in A, in B, or in both A and B Intersection(A^B) contains all outcomes that are in both A and B, but not in A alone or B alone. General Addition Rule: P(AUB) = P(A) + P(B) P(A^B) Multiplication Rule: P(A^B) = P(A)P(B A) Conditional Probability P(B A) of an event B, given that event A has occurred: P(B A) = P(A^B)/P(A) when P(A) > 0 If A and B are disjoint (mutually exclusive) then P(A^B) = 0 and P(AUB) = P(A) + P(B) A and B are independent when P(B A) = P(B) Venn diagram or tree diagrams useful for organization.