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Probability and Counting Rules

In probability theory and statistics, a probability distribution is a way of describing the probability of an event, or the possible outcomes of an experiment, in a given state of the world. The set of all possible outcomes of the experiment (the sample space) is a subset of the sample space of all possible states of the world (the state space). For example, the probability that a fair coin comes up heads is 0.5, or half of the 2 possible outcomes of a coin flip (heads or tails). In the state space, the event "heads" corresponds to the outcome "heads" (followed by a coin flip), and "tails" corresponds to the outcome "tails" (followed by a coin flip). Similarly, the event "tail" corresponds to the outcome "tails" (followed by a coin flip), and "heads" corresponds to the outcome "heads" (followed by a coin flip). In the state space, "tails" and "heads" are the two possible outcomes of a coin flip, and the event of "tails" corresponds to the outcome "tails" (followed by a coin flip), and the event of "heads" corresponds to the outcome "heads" (followed by a coin flip). In probability theory and statistics, the probability distribution of a random variable (a function whose domain is the sample space) is a table that lists the probabilities of the values of the random variable. The probability distribution of a discrete random variable, such as the number of heads in a coin toss, is a table that lists the probabilities of the outcomes in the sample space, with each outcome in the sample space as the row label and the outcome of the random variable as the column label. In probability theory and statistics, the probability distribution of a continuous random variable is a table that lists the probabilities of the values of the random variable on a range of values; for example, the probability distribution of a continuous random variable X with a probability density function "f"("x") is a table that lists the probabilities of the values of X on the interval [0, 1] . In probability theory and statistics, a probability density function (also called a probability function or probability density) is a function that describes the probability of the value of another random variable, called the "dependent" variable, given the value of the random variable describing the probability, called the "independent" variable. For example, the probability density function of the random variable X is the function that gives the probability that X is in a given interval (the interval "I") when the value of X is given by a random variable Y. For example, if the probability of X is 0.5, and the probability of Y is 0.5, then the probability density function of X is 0.5 at x = 0 and 0.5 at x = 1. The probability distribution of the random variable X is its probability density function; that is, its probability distribution is the function F(x) = P(X = x). Similarly, the probability distribution of the random variable Y is the function G(y) = P(Y = y). The probability distribution of X is the function F(x) to which G(y) is the inverse function. The probabilities of the outcomes of an experiment are often given as the frequencies of the outcomes in the sample space. For example, the probability that a fair coin comes up heads is half the probability that a coin will land heads up. The probabilities of the outcomes of an experiment are often given as the relative frequencies of the outcomes in the sample space. For example, the probability that a fair coin comes up heads is twice the frequency of a coin landing heads up. In probability theory and statistics, the probability distribution of a random variable is the probability density function of the random variable. The probability distribution of a discrete random variable is the probability table of the random variable. The probability distribution of a continuous random variable is the probability function of the random variable. In probability theory and statistics, a random variable or stochastic variable is a function whose domain is the sample space of a discrete or continuous random experiment. For example, the sample space of a fair coin toss is the set of all possible outcomes, which can be described by the following three-element set: "H" = {0, 1, 2}. The probability distribution of a random variable is the probability table of the random variable. The probability distribution of a random variable is the probability mass function of the random variable. The probability mass function is the probability distribution of a random variable given its probability distribution. The probability mass function is often denoted by the Greek letter ? (pi), the first letter of the word probability. For example, the sample space of a fair coin toss is the three-element set {

Sample Spaces and Probability

5 Practice Problems
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03:11
Mathematical Statistics with Applications

According to the Washington Post, nearly 45\% of all Americans are born with brown eyes, although their eyes don't necessarily stay brown. $^{\star}$ A random sample of 80 adults found 32 with brown eyes. Is there sufficient evidence at the .01 level to indicate that the proportion of brown eyed adults differs from the proportion of Americans who are born with brown eyes?

Hypothesis Testing
Common Large-Sample Tests
03:03
Mathematical Statistics with Applications

The hourly wages in a particular industry are normally distributed with mean $\$ 13.20$ and standard deviation $\$ 2.50 .$ A company in this industry employs 40 workers, paying them an average of $\$ 12.20$ per hour. Can this company be accused of paying substandard wages? Use an $\alpha=.01$ level test.

Hypothesis Testing
Common Large-Sample Tests
01:00
Elementary Statistics a Step by Step Approach

Population of Hawaii The population of Hawaii is $22.7 \%$ white, $1.5 \%$ African-American, $37.7 \%$ Asian, $0.2 \%$ Native American/Alaskan, $9.46 \%$ Native Hawaiian/Pacific Islander, $8.9 \%$ Hispanic, $19.4 \%$ two or more races, and $0.14 \%$ some other. Choose one Hawaiian resident at random. What is the probability that he/she is a Native Hawaiian or Pacific Islander? Asian? White?

Probability and Counting Rules
Sample Spaces and Probability
Richard Miller

The Addition Rules for Probability

11 Practice Problems
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04:50
Elementary Statistics

Probability of a Run of Four Use a simulation approach to find the probability that when six consecutive babies are born, there is a run of at least four babies of the same sex. Describe the simulation procedure used, and determine whether such runs are unlikely.

Probability
Probabilities Through Simulations
Willis James
04:35
Elementary Statistics

Describe the simulation procedure. (For example, to simulate 10 births, use a random number generator to generate 10 integers between 0 and 1 inclusive, and consider 0 to be a male and 1 to be a female.)
Lefties Ten percent of people are left-handed. In a study of dexterity, 15 people are randomly selected. Describe a procedure for using software or a TI- $83 / 84$ Plus calculator to simulate the random selection of 15 people. Each of the 15 outcomes should be an indication of one of two results: (1) Subject is left-handed; (2) subject is not left-handed.

Probability
Probabilities Through Simulations
Oluwadamilola Ameobi
02:56
Elementary Statistics

Simulating Coin Flips One student conducted the simulation described in Example 3 and stated that the probability of getting a sequence of six 0s or six 1s is 0.977. What is wrong with that statement?

Probability
Probabilities Through Simulations
Oluwadamilola Ameobi

The Multiplication Rules and Conditional Probability

22 Practice Problems
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02:01
Elementary Statistics

System Reliability Refer to the accompanying figure in which surge protectors p and $q$ are used to protect an expensive high-definition television. If there is a surge in the voltage. the surge protector reduces it to a safe level. Assume that each surge protector has a 0.99 probability of working correctly when a voltage surge occurs.
a. If the two surge protectors are arranged in series, what is the probability that a voltage surge will not damage the television? (Do not round the answer)
b. If the two surge protectors are arranged in parallel, what is the probability that a voltage surge will not damage the television? (Do not round the answer)
c. Which arrangement should be used for the better protection?
(FIGURE CANNOT COPY)

Probability
Multiplication Rule: Basics
Christopher Stanley
02:20
Elementary Statistics

Roller Coaster The Rock 'n' Roller Coaster at Disney-MGM Studios in Orlando has 2 scats in each of 12 rows. Riders are assigned to seats in the order that they arrive. If you ride this roller coaster once, what is the probability of getting the coveted first row? How many times must you ride in order to have at least a $95 \%$ chance of getting a first-row scat at least once?

Probability
Multiplication Rule: Complements and Conditional Probability
Joseph Russell
01:22
Elementary Statistics

Redundancy in Alarm Clocks A statistics student wants to ensure that she is not late for an early statistics class because of a malfunctioning alarm clock. Instead of using one alarm dock, she decides to use three. What is the probability that at least one of her alarm clocks works correctly if each individual alarm clock has a $90 \%$ chance of working correctly? Does the student really gain much by using three alarm clocks instead of only one? How are the results affected if all of the alarm clocks run on electricity instead of batteries?

Probability
Multiplication Rule: Complements and Conditional Probability
Kari Hasz

Counting Rules

17 Practice Problems
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01:21
Elementary Statistics

As of this writing, the Powerball lottery is run in 29 states. Winning the jackpot requires that you select the correct five numbers between 1 and 55 and, in a separate drawing, you must also select the correct single number between 1 and $42 .$ Find the probability of winning the jackpot.

Probability
Counting
Jeff Vermeire
02:25
Elementary Statistics

ATM Machine You want to obtain cash by using an ATM machine, but it's dark and you anit see your card when you insert it. The card must be inserted with the front side up and the printing configured so that the beginning of your name enters first.
a. What is the probability of selecting a random position and inserting the card, with the result that the card is inserted correctly?
b. What is the probability of randomly selecting the card's position and finding that it is incorrectly inserted on the first attempt, but it is correctly inserted on the second attempt?
c. How many random selections are required to be absolutely sure that the card works because
it is inserted correctly?

Probability
Counting
Jeff Vermeire
00:35
Elementary Statistics

DNA (dœoxyribonucleic acid) is made of nucleotides. Each nudeotide can contain any one of these nitrogenous bases: $A$ (adenine), $G$ (guanine), $C$ (cytosinc). T (thymine). If one of those four bases (A, G. C. T) must be sciected three times to form a linear triplet, how many different triplets are possible? Note that all four bases can be sclected for cach of the three components of the triplet.

Probability
Counting
Jeff Vermeire

Probability and Counting Rules

28 Practice Problems
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02:53
Elementary Statistics

Can computers "think"? According to the living tort, a computer an be considered to think if, when a person communicates with it, the person belicues he or she is communicating with another person instead of a computer. In an experiment at Boston's Computer Museum, each of 10 judges communicated with four computers and four other people and was asked to distinguish between them.
a. Assume that the first judge cannot distinguish between the four computers and the four people. If this judge makes random guesses, what is the probability of correctly identifying the four computers and the four people?
b. Assume that all 10 judges cannot distinguish between computers and people, so they make random guesses. Based on the result from part (a), what is the probability that all 10 judges make all correct pueses? (That event would lead us to conclude that computers cannot Think" when, according to the Turing test, they an.)

Probability
Counting
Jeff Vermeire
02:41
Elementary Statistics

Variable Names A common computer programming rule is that names of variables must be between 1 and 8 characters long. The first character can be any of the 26 letters, while successive characters an be any of the 26 letters or any of the 10 digits. For example, allowable variable names are $A, B B B$, and $M 3477 K$. How many different variable names are possible?

Probability
Counting
Jeff Vermeire
00:51
Elementary Statistics

In Phase I of a clinical trial with gene therapy used for treating HIV, five subjects were treated (based on data from Medical News Today ). If 20 people were eligible for the Phase I treatment and a simple random sample of five is selected, how many different simple random samples are possible? What is the probability of each simple random ample?

Probability
Counting
Jeff Vermeire

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