Statistics chapter 5, 6
The notation __________ is used for the probability of sucess on any trial in a binomial experiment.
p
Phrase Math Symbol
Phrase Math Symbol "at least" or "no less than" or "greater than or equal to" ≥ "more than" or "greater than" > "fewer than" or "less than" < "no more than" or "at most" or "less than or equal to ≤ "exactly" or "equals" or "is" =
Probability
Probability is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the long-term proportion with which a certain outcome will occur in situations with short-term uncertainty. Use the probability applet to simulate flipping a coin 100 times. Plot the proportion of heads against the number of flips. Repeat the simulation.
certainty
If an event is a certainty, the probability of the event is 1.
EXAMPLE A Probability Model
In a bag of peanut M&M milk chocolate candies, the colors of the candies can be brown, yellow, red, blue, orange, or green. Suppose that a candy is randomly selected from a bag. The table shows each color and the probability of drawing that color. Verify this is a probability model. Color Probability Brown 0.12 Yellow 0.15 Red 0.12 Blue 0.23 Orange 0.23 Green 0.15 • All probabilities are between 0 and 1, inclusive. • Because 0.12 + 0.15 + 0.12 + 0.23 + 0.23 + 0.15 = 1, rule 2 (the sum of all probabilities must equal 1) is satisfied.
experiment
In probability, an experiment is any process that can be repeated in which the results are uncertain.
Apply the Rules of Probabilities
Rules of probabilities 1. The probability of any event E, P(E), must be greater than or equal to 0 and less than or equal to 1. That is, 0 ≤ P(E) ≤ 1. 2. The sum of the probabilities of all outcomes must equal 1. That is, if the sample space S = {e1, e2, ..., en}, then P(e1) + P(e2) + ... + P(en) = 1
If an experiment has n equally likely outcomes and if the number of ways that an event E can occur is m, then the probability of E, P(E), is P(E)= number of ways that E can occur / number of possible outcomes =m/n
So, if S is the sample space of this experiment, then P(E)=N(E)/N(S) where N(E) is the number of outcomes in E, and N(S) is the number of outcomes in the sample space.
Standard Deviation of a Discrete Random Variable
Standard Deviation of a Discrete Random Variable The standard deviation of a discrete random variable X is given by where x is the value of the random variable, μX is the mean of the random variable, and P(x) is the probability of observing a value of the random variable.
Surveys are probability experiments
Surveys are probability experiments. Why? Each time a survey is conducted, a different random sample of individuals is selected. Therefore, the results of a survey are likely to be different each time the survey is conducted because different people are included.
Probability of outcome
The long-term proportion in which a certain outcome is observed is the probability of that outcome.
sample space, S,
The sample space, S, of a probability experiment is the collection of all possible outcomes for that experiment.
Sample space s
The sample space, S, of a probability experiment is the collection of all possible outcomes. The set of all possible outcomes of an experiment is called the SAMPLE SPACE of the experiment.
What does it mean to say that the trials in a binomial experiment are independent of each other? The outcome of one trial does not affect the outcomes of the other trials.
Trials are independent if the outcome of any trial does not affect the outcomes of the other trials. This is a requirement for a binomial experiment. For example, if a card is drawn from a standard deck of cards and replaced before the next draw, then the cards drawn are independent of each other.
Two events E and F are INDEPENDENT if the occurrence of event E in a probability experiment does not affect the probability of event F.
Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F.
discrete random variable
discrete random variable has either a finite or countable number of values. The values of a discrete random variable can be plotted on a number line with space between each point.
empirical evidence
empirical evidence, that is, evidence based on the outcomes of a probability experiment.
A continuous random variable
A continuous random variable has infinitely many values. The values of a continuous random variable can be plotted on a line in an uninterrupted fashion.
Expected value
A mathematical way to use probability to determine what to expect over the long run is called Expected value
which numbers COULD be used to represent the probability of an event. (The definition of probability states the probability of an event occuring must be contained in the interval [0,1] or [0%,100%].)
Zero, 320/1058, 33.3%, 0.0002
probability distribution
A probability distribution provides the possible values of the random variable X and their corresponding probabilities. A probability distribution can be in the form of a table, graph or mathematical formula. P(x) Rules for a Discrete Probability Distribution Let P(x) denote the probability that the random variable X equals x; then 1. Σ P(x) = 1 2. 0 ≤ P(x) ≤ 1
What is the difference between an outcome and an event? An outcome is the result of a single probability experiment. An event is a set of one or more possible outcomes.
A probability experiment is an action, or trial, through which specific results (counts, measurements, or responses) are obtained. The result of a single trial in a probability experiment is an outcome. The set of all possible outcomes of a probability experiment is the sample space. An event is a subset of the sample space. It may consist of one or more outcomes.
probability histogram
A probability histogram is a histogram in which the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of that value of the random variable.
probability model
A probability model lists the possible outcomes of a probability experiment and each outcome's probability.
probability model
A probability model lists the possible outcomes of a probability experiment and each outcome's probability. A probability model must satisfy rules 1 and 2 of the rules of probabilities.
random variable
A random variable is a numerical measure of the outcome from a probability experiment, so its value is determined by chance. Random variables are denoted using letters such as X.
What is a random variable?
A random variable is a numerical measure of the outcome of a probability experiment.
definition of a continuous random variable
A random variable is a numerical measure, having values that can be plotted on a line in an uninterrupted fashion, of the outcome of a probability experiment.
Event E
An event is any collection of outcomes from a probability experiment. An event may consist of one outcome or more than one outcome. We will denote events with one outcome, sometimes called simple events, ei. In general, events are denoted using capital letters such as E.
Using Relative Frequencies to Approximate Probabilities
An insurance agent currently insures 182 teenage drivers (ages 16 to 19). Last year, 24 of the teenagers had to file a claim on their auto policy. Based on these results, the probability that a teenager will file a claim on his or her auto policy in a given year is 24/182~0.132 So, for every 100 insured teenage drivers, we expect about 13 to have a claim on their auto policy.
unusual event
An unusual event is an event that has a low probability of occurring. usually less than .5 Statisticians typically use cutoff points of 0.01, 0.05, 0.01, 0.05, and 0.10.
Objective 2 •Compute and Interpret Probabilities Using the Empirical Method
Approximating Probabilities Using the Empirical Approach The probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the experiment. P(E) ≈ relative frequency = Frequency of E/# of trials of experiments
expected value
Because the mean of a random variable represents what we would expect to happen in the long run, it is also called the expected value, E(X), of the random variable.
Complement Rule
If E is an event, then the Complement Rule can be stated as: P(event) = 1−P(event does NOT occur) or P(not E) = 1−P(E) . The probability of the complement of an event is one MINUS the probability of the event itself. Since either the event or the complement must occur, together their probabilities must add to one. ex:Find the probability P(not E) if P(E)=0.25. The probability P(not E) is . 75
EXAMPLE Identifying Events and the Sample Space of a Probability Experiment
Consider the probability experiment of having two children. (a) Identify the outcomes of the probability experiment. (b) Determine the sample space. (c) Define the event E = "have one boy". EXAMPLE Identifying Events and the Sample Space of a Probability Experiment (a) e1 = boy, boy, e2 = boy, girl, e3 = girl, boy, e4 = girl, girl (b) {(boy, boy), (boy, girl), (girl, boy), (girl, girl)} (c) {(boy, girl), (girl, boy)}
Criteria for a Binomial Probability Experiment
Criteria for a Binomial Probability Experiment An experiment is said to be a binomial experiment if 1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial. 2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials. 3. For each trial, there are two mutually exclusive (or disjoint) outcomes, success or failure. 4. The probability of success is fixed for each trial of the experiment.
DEFINITION A subjective probability
DEFINITION A subjective probability is a probability that is determined based on personal judgment.
The binomial probability distribution is a __________ probability distribution that describes probabilities for experiments in which there are two __________ outcomes.
Discrete, mutually exclusive
Empirical probability is used
Empirical probability is used when probabilities cannot be theoretically calculated. For example, life insurance companies use empirical probabilities to determine the chance of an individual in a certain profession, with certain risk factors, living to age 65.
fixed probability of success,
For a fixed probability of success, p, as the number of trials n in a binomial experiment increase, the probability distribution of the random variable X becomes bell-shaped. As a general rule of thumb, if np(1 - p) > 10, then the probability distribution will be approximately bell-shaped.
The classical probability (or theoretical probability) of event E, denoted by ______, is the ______ divided by ______. P(E), number of outcomes in E total number of possible outcomes.
If an event E has n(E) equally likely outcomes and its sample space S has n(S) equally likely outcomes, the theoretical probability of event E, denoted by P(E), is shown below.
impossible
If an event is impossible, the probability of the event is 0.
The Multiplication Rule P(E and F)=P(E)•P(F) applies only to which type of events? Independent
If two events are independent, it means that the probability of one is not affected by the occurrence of the other. In this case, we can find the probability of both occurring by simply multiplying the individual probabilities.
What do we call the outcome of zero?
Impossible event
Which probability method requires that an experiment have equally likely outcomes? Classical
In classical probability, the probability of an event is found by calculating the ratio of the number of ways the event can occur to the number of outcomes in the sample space. This is only valid if all the outcomes in the sample space are equally likely.
simple random sampling
In simple random sampling, each individual has the same chance of being selected. Therefore, we can use the classical method to compute the probability of obtaining a specific sample.
P(E)
In the following probability rules, the notation P(E) means "the probability that event EE occurs.
Interpretation of the Mean of a Discrete Random Variable
Interpretation of the Mean of a Discrete Random Variable Suppose an experiment is repeated n independent times and the value of the random variable X is recorded. As the number of repetitions of the experiment increases, the mean value of the n trials will approach μX, the mean of the random variable X. In other words, let x1 be the value of the random variable X after the first experiment, x2 be the value of the random variable X after the second experiment, and so on. Then The difference between and μX gets closer to 0 as n increases.
Determine if the following probability experiment represents a binomial experiment. A random sample of 80 professional athletes is obtained, and the individuals selected are asked to state their heights.
No, this probability experiment does not represent a binomial experiment because the variable is continuous, and there are not two mutually exclusive outcomes.
Symbols / Notation Used in the Binomial Probability Distribution
Notation Used in the Binomial Probability Distribution •There are n independent trials of the experiment. •Let p denote the probability of success so that 1 - p is the probability of failure. •Let X be a binomial random variable that denotes the number of successes in n independent trials of the experiment. So, 0 < x < n.
The Law of Large Numbers
The Law of Large Numbers As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome.
Compute and Interpret the Mean of a Discrete Random Variable
The Mean of a Discrete Random Variable The mean of a discrete random variable is given by the formula where x is the value of the random variable and P(x) is the probability of observing the value x.
The word "and" in probability implies that we should use the Multiplication Rule.
The Multiplication Rule is used to find the probability that events E and F both occur. The events must be independent to use this rule.
The probability of an event E
The [P(E)] probability of an event E occurring is approximately the number of times event E is observed divided by the number of repetitions (or trials) of the experiment.
The empirical method gives an approximate probability of an event by conducting a probability experiment.
The classical method of computing probabilities does not require that a probability experiment actually be performed. Rather, it relies on counting techniques. The classical method of computing probabilities requires equally likely outcomes. An experiment has equally likely outcomes when each outcome has the same probability of occurring.
Which of the following statements correctly describes the complement of event E? The complement of event E is the set of outcomes which are in the sample space but not in event E.
The complement of an event is the opposite of the event. If you know the probability of an event, you can find the probability of its complement by subtracting the probability of the event from 1.
The mean (or expected value)
The mean (or expected value) of a random variable represents what we would expect to happen, on average, in the long run. It is the average value of the random variable for a procedure repeated an infinite number of times. For a discrete probability distribution, the mean will not necessarily be a possible value of the random variable though. Next Question
Independent Events
Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F. If two events E and F are independent, then the P(E and F)= P(E)xP(F)
Disjoint or Mutually Exclusive Events
Two events are disjoint if they have no outcomes in common. Another name for disjoint events is mutually exclusive events. If E and F are disjoint events, then the P(E o F)=P(E)+P(F)
EXAMPLE Identifying Binomial Experiments
Which of the following are binomial experiments? (a) A player rolls a pair of fair die 10 times. The number X of 7's rolled is recorded. Binomial experiment
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. An experimental drug is administered to 170170 randomly selected individuals, with the number of individuals responding favorably recorded.
Yes, because the experiment satisfies all the criteria for a binomial experiment.
EXAMPLE Identifying Binomial Experiments
b) The 11 largest airlines had an on-time percentage of 84.7% in November, 2001 according to the Air Travel Consumer Report. In order to assess reasons for delays, an official with the FAA randomly selects flights until she finds 10 that were not on time. The number of flights X that need to be selected is recorded. Not a binomial experiment - not a fixed number of trials.
binomial probability distribution
binomial probability distribution is a discrete probability distribution that describes probabilities for experiments in which there are two mutually exclusive (disjoint) outcomes. These two outcomes are generally referred to as success (such as making a free throw) and failure (such as missing a free throw). Experiments in which only two outcomes are possible are referred to as binomial experiments, provided that certain criteria are met.
EXAMPLE Identifying Binomial Experiments
c) In a class of 30 students, 55% are female. The instructor randomly selects 4 students. The number X of females selected is recorded. Not a binomial experiment - the trials are not independent.
Determine which numbers COULD NOT be used to represent the probability of an event. (The definition of probability states the probability of an event occuring must be contained in the interval [0,1] or [0%,100%].)
−1.5, because probability values cannot be less than 0. 64/25, because probability values cannot be greater than 1.
