Chapter 4
Complement Rule
says that the probability of the complement of an event is 1 minus the probability of the event. P(Ac) = 1 - P(A)
A store owner reports that the probability that a customer who purchases a lawn mower will also purchase an extended warranty is 0.68. How do we write an interpretation sentence of the probability 0.68 ?
All customer: long-run 68%: Relative Frequency Extended Warranty: Of an Event
Geometric Probability Formula
P(y=k) = (1-p)^k-1 (p) EX: Assume 18% of people are left handed. If we select 9 people at random, find the probability that the first lefty is the fifth person selected. *(0.82)4 (0.18) = 0.081
Patterns in data do not necessarily mean that _______________
variation is not random
parameter
-A numerical value measuring a characteristic of a population or the distribution of a random variable -single, fixed value. -two parameters we are interested in when studying random variables are μ and σ. -should be interpreted using appropriate units and within the context of a specific population. So, don't just find the expected value - but tell what it means in the context of the problem!
Tools for producing random outcomes
-Coins, dice, or playing cards -Spinners -Random number generators -Random digit tables
Complementary Events
-In probability theory, the complement of an event A is the event "not A". -This complementary event is often denoted A^c. EX: If our event A is "it rains today," then the complement, Ac, is the event "it doesn't rain today." If you're drawing a card from a standard 52-card deck, and the event A is "you draw a diamond," then the complement Ac is "you don't draw a diamond." *What is common to these examples is that the event A and its complement Ac are mutually exclusive and exhaustive. They are mutually exclusive because the two events cannot occur at the same time, and they are exhaustive because the sum of their probabilities must add to 100%.
Law of Large #'s
-States that simulated probabilities tend to get closer to true probability as the number of trials increases. *The idea that the relative frequency of an event will converge on the probability of the event, as the number of trials increases, i
Something is considered to be random if:
-We can't predict exactly what will happen next. -In the long run, we know about what to expect
7 Steps for a Simulation
1.) Identify the component to be repeated 2.) Explain how you will model the outcome 3.) Explain how you will simulate the trial 4.) State what the response variable is 5.) Run Several tests 6.) Analyze the response variable 7.) State your conclusion 1&7 are Real World. 2-6 are Model
Basic Probability Rules
1.) The probability of any event E is a number between 0 and 1, inclusive. This is denoted by 0 ≤ P(E) ≤ 1 2.) If an event E cannot occur, then its probability is zero. 3.) If an event E is certain, then its probability is 1. 4.) The sum of the probabilities of all the outcomes in the sample space is 1.
Binomial, Geometric, Neither?
Assume 18% of people are left handed. If we select 9 people at random, find the following probabilities: -The first lefty is the fifth person. Geometric -There are exactly four lefties in the group. Binomial -There is at least one lefty in the group. Binomial -The first lefty is the sixth or seventh person.Geometric
To determine whether a variable is "binomial" or not, ask yourself these questions:
B: Is it binary? (only 2 outcomes) I: Are the trials independent? (probability of success does not change) N: Is the number of trials fixed? S: Are we counting the number of successes?
Independent Events
Events A and B are independent if, and only if, knowing whether A has occurred (or will occur) does not change the probability that event B will occur. P(A|B)=P(A) P(B|A)=P(B)
Binomial Probability Formula
EX: For a certain drug, 15% of the people who take it develop side effects. Suppose 9 people have taken the drug. Find the probability that exactly three develop side effects. ( 9 3) (.15^3)(.85^6)= 107
Are these scenerios independent?
EXAMPLE 1: Consider the experiment where three marbles are drawn without replacement from a bag containing 20 red and 40 blue marbles, and the number of red marbles drawn is recorded. Is this a binomial experiment? No! The key here is the lack of independence - since the marbles are drawn without replacement, the marble drawn on the first will affect the probability of later marbles. EXAMPLE 2: A fair six-sided die is rolled ten times, and the number of 6's is recorded. Is this a binomial experiment? Yes! There are fixed number of trials (ten rolls), each roll is independent of the others, there are only two outcomes (either it's a 6 or it isn't), and the probability of rolling a 6 is constant.
Mutually exclusive
Events that cannot occur at the same time.
How to determine if a variable is geometric
F: Are you interested in when the first success occurs? I: Are the trials independent? (probability of success does not change) B: Is it binary? (only 2 outcomes)
a geometric random variable
For a sequence of independent trials, X, gives the number of the trial on which the first success occurs. Each trial has two possible outcomes (success or failure) with the probability of success p and the probability of failure 1 - p.
Mean & Standard Deviation of a binomial random variable
If a count X of successes has the binomial distribution with number of trials "n" and probability of success "p", the mean and standard deviation of X EX: A survey in USA Today found that 30% of pet owners agreed with the statement, "I usually pay to have my pet bathed professionally rather than do it myself." In a random sample of 150 pet owners, how many would you expect to agree with the statement? With what standard deviation? μ = np = (150)(0.30) = 45 pet owners σ = sqrt (150)(0.30)(0.70) = 5.612 pet owners
Geometric Parameters
If a random variable is geometric, its mean, ux, is 1/p and its standard deviation, σx, is (sqrt root (1-p))/ p EX: A survey conducted by the Harris polling organization discovered that 70% of all Americans are overweight. Suppose that a number of randomly selected Americans are weighed. How many Americans would you expect to weigh before you encounter the first overweight individual? With what standard deviation? mean= 1/.7- 1.43 Standard deviation= sqrt root (.3)/ (.7) = .78
If the outcomes in the sample space are equally likely, then the probability an event E will occur is defined as the fraction:....
Number in outcomes in Event E/ Total number of outcomes in sample space
Interpreting Binomial Distributions
Probabilities and parameters (μ,σ) for a binomial distribution should be interpreted using appropriate units and within the context of a specific population of situation. EX: A company that ships crystal bowls claims that bowls arrive undamaged in 95 percent of the shipments. Let the random variable G represent the number of shipments with undamaged bowls in 25 randomly selected shipments. Random variable G follows a binomial distribution with a mean of 23.75 shipments and a standard deviation of approximately 1.09 shipments. Interpret the mean in this context. For all possible shipments of size 25, the average number of undamaged shipments is equal to 23.75.
Interpreting Geometric Distributions
Probabilities and parameters (μ,σ) for a geometric distribution should be interpreted using appropriate units and within the context of a specific population of situation. EX:Let W represent the number of attempted experiments to get one experiment that is not successful. The random variable W has a geometric distribution with mean 4 and standard deviation 3.5. How would you best interpret the standard deviation in this context? Values of W typically vary by about 3.5 attempted experiments, on average, from the mean of 4 experiments.
Multipication Rule
Probability that A and B both will occur is equal to the probability that event A will occur multiplied by the probability that event B will occur, given that A has occured.
Why use a simulation?
Some situations do not lend themselves to precise mathematical treatment. Others may be difficult, time-consuming, or expensive to analyze. In these situations, simulation may approximate real-world results; yet, require less time, effort, and/or money than other approaches.
Joint Probability
The probability that events A and B both occur -It is denoted P(A ∩ B). -Joint probability is associated with the word "and". If you were asked to "Find the probability that a student is female and plays soccer", you would be finding joint probability.
mutually exclusive (disjoint)
Two events that cannot happen at the same time *P(A ∩ B) = 0
probability distribution
a description of some chance process that consists of two parts: -a sample space S and -a probability for each outcome. -can be represented as a graph, a table, or a function showing the probabilities associated with values of a random variable. -We have defined the variable X = the number of heads obtained when a coin is tossed three times. The histogram at right displays the same information as the table below. Notice that it is symmetrical and unimodal. We could use SOCS to describe this distribution.
random variable
a variable whose numerical value is based on the outcome of a random event -For example, Let X = the result of tossing a die. Then we might find X1 =6, X2 = 3, X3 = 1, and so on. -Another example: Let Y = the height (in inches) of a person selected at random. *Notice that variable X is discrete and variable Y is continuous. -A discrete random variable is a variable that can only take a countable number of values. Each value has a probability associated with it. The sum of the probabilities over all of the possible values must be 1.
Simulation
a way to model random events, such that simulated outcomes closely match real-world outcomes .*By observing simulated outcomes, researchers gain insight on the real world.
Event
any collection of outcomes from some chance process. *an event is a subset of the sample space. *Events are usually designated by capital letters, like A, B, C, and so on. *For example: When rolling a die, the sample space of possible outcomes is S = {1, 2, 3, 4, 5, 6}. We might define event A = die roll is odd. The elements of the sample space S that fit this event are A = {1, 3, 5}. The probability of the event A, written as P(A) is the 3/6 or 1⁄2. So we would write P(A) = 0.5.
Cumulative Probability Distribution
can be represented as a table or function showing the probability of being less than or equal to each value of the random variable. -To represent a cumulative probability distribution graphically, we would use an ogive.
binomial random variable
counts the number of successes in n repeated independent trials, each trial having two possible outcomes (success or failure), with the probability of success p and the probability of failure 1 - p.
The relative frequency of an outcome or event in simulated or empirical data can be used to estimate ________________
probability of that outcome or event. EX: Long run:, a merchant notices one day that 5 out of 50 visitors to her store make a purchase. The next day, 20 out of 50 visitors make a purchase. The two relative frequencies (5/50 or 0.10 and 20/50 or 0.40) differ. However, summing results over many visitors, she might find that the probability that a visitor makes a purchase gets closer and closer 0.20.
The Addition Rule
states that the probability that event A or event B or both will occur is equal to the probability that event A will occur plus the probability that event B will occur, minus the probability that both events will occur. This is denoted: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
The Multiplication Rule
states that the probability that events A and B will both occur is equal to the probability that event A will occur multiplied by the probability that event B will occur, given that event A has occurred. This is denoted P(A ∩ B) = P(A)∙P(B|A)
What is probability
the long-run relative frequency of an event.
Conditional Probability
the probability of an event ( A ), given that another ( B ) has already occurred.
sample space (S)
the set of all possible non-overlapping outcomes. *Outcomes are considered equally likely if they have the same chance of occurring. -A family has three children. Give the sample space for the genders of the children. Are the outcomes equally likely? S = {BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG} Yes - outcomes are equally likely -A family has three children. Give the sample space for the number of boys in the family. Are the outcomes equally likely? S = {0, 1, 2, 3} Not equally likely. There is only one way to have 3 boys (BBB) but there are three different ways to have two boys (BBG, BGB, GBB)
Intersection of two sets (A ∩ B)
the set that contains all elements of A that also belong to B. EX: Given A = {1, 3, 5, 7, 9, 11, 13, 15} and B = {0, 3, 6, 9, 12, 15} Then (A ∩ B) = {3, 9, 15}
Union of two sets (A ∪ B)
the set that contains all elements that belong to set A or set B or both sets A and B. -For example: Given A = {1, 3, 5, 7, 9, 11, 13, 15} and B = {0, 3, 6, 9, 12, 15} Then (A ∪ B) = {0, 1, 3, 5, 6, 7, 9, 11, 12, 13, 15} -The probability that event A or event B (or both) will occur is denoted P(A ∪ B). -When probability questions use the word "or"it is always implied that it means one or the other or both. If you were asked to "Find the probability that a student is female or plays soccer", you would need to find the probability that a student is female, or plays soccer, or is a female soccer player.
Exhaustive
the sum of their probabilities must add to 100%.
If necessary, you can use a ___________ to determine the sample space
tree diagram
To know if the outcomes of an event are "unusual", we need to know what ____________and the ________________
typically occurs and the amount of variability.
