Test 2

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binomial experiment

1. The experiment is repeated for a fixed number of​ trials, where each trial is independent of the other trials. 2. There are only two possible outcomes of interest for each trial. The outcomes can be classified as a success​ (S) or as a failure​ (F). 3. The probability of a success​ P(S) is the same for each trial. 4. The random variable x counts the number of successful trials.

Use the Addition Rule: You select a card from a standard deck of 52 playing cards. Find the probability that the card is a 4 or an ace.

A card that is a 4 cannot be an ace. So the events are mutually exclusive. P (4 or ace) = P(4) + P(ace) = 4/52 + 4/52 = 8/52 = 2/13 =.154

Multiplication rule

A rule of probability stating that the probability of two or more independent events occurring together can be determined by multiplying their individual probabilities.

When is an event unusual?

An event is considered unusual if its probability is less than or equal to 0.05

Describe the difference between the value of x in a binomial distribution and in a geometric distribution.

In a binomial​ distribution, the value of x represents the number of successes in n​ trials, while in a geometric​ distribution, the value of x represents the first trial that results in a success.

What is the significance of the mean of a probability​ distribution?

It is the expected value of a discrete random variable.

What is the difference between independent and dependent​ events?

Two events are independent when the occurrence of one event does not affect the probability of the occurrence of the other event. Two events are dependent when the occurrence of one event affects the probability of the occurrence of the other event.

What is a discrete probability​ distribution? What are the two conditions that determine a probability​ distribution?

A discrete probability distribution lists each possible value a random variable can​ assume, together with its probability. The probability of each value of the discrete random variable is between 0 and​ 1, inclusive, and the sum of all the probabilities is 1.

Can two events with nonzero probabilities be both independent and mutually​ exclusive?

No, two events with nonzero probabilities cannot be independent and mutually exclusive because if two events are mutually​ exclusive, then when one of them​ occurs, the probability of the other must be zero.

An example of dependent events

When drawing two cards​ (without replacement) from a standard​ deck, the outcome of the second draw is dependent on the outcome of the first draw.

When you calculate the number of combinations of *r* objects taken from a group of *n* objects what are you​ counting? Give an example

You are counting the number of ways to select r of the n objects without regard to order. An example of a combination is the number of ways a group of teams can be selected for a tournament

The Addition Rule (In words)

In words, to find the probability that one event or the other will occur, add the individual probabilities of each event and subtract the probability that they both occur.

Is the expected value of the probability distribution of a random variable always one of the possible values of​ x? Explain.

No, because the expected value may not be a possible value of x for one​ trial, but it represents the average value of x over a large number of trials.

What is the total area under the normal​ curve?

1

Construct probability distribution steps

1) Find the frequency of the data. The sum total of all the variables. 2) Divide the individual variables by the frequency to find the probability of the outcome.

In most​ applications, continuous random variables represent counted​ data, while discrete random variables represent measured data. T/F

False. In most​ applications, discrete random variables represent counted​ data, while continuous random variables represent measured data.

If two events are mutually​ exclusive, why is P (A and B ) = 0​?

Because A and B cannot occur at the same time

Determine if mutually exclusive: Event A: Randomly select a blood donor with type O blood. Event B: Randomly select a female blood donor

Because the donor can be a female with type O blood, the events are not mutually exclusive.

Determine if mutually exclusive: Event A: Randomly select a male student. Event B: Randomly select a nursing major.

Because the student can be a a male nursing major, the events are not mutually exclusive.

Decide whether the random variable x is discrete or continuous. Explain your reasoning. Let x represent the volume of blood drawn for a blood test.

Continuous, because x is a random variable that cannot be counted

T/F If two events are​ independent, ​P(A|B) = P(B)

False; if events A and B are​ independent, then​ P(A and ​B)= ​P(A) * ​P(B).

Sales Volume | Months 0k-24,999 | 3 25k-49,999 | 5 50k-74,999 | 6 75k -99,999 | 7 100k-124,999 | 9 125k-149,999 | 2 150k-179,999 | 3 175k-199,999 | 1 Using the sales pattern, find the probablity that the sales rep will sell between 75k - 124,999 next month

Define events A and B A = {monthly sales between 75k-99,999} and B = {monthly sales between 100k and 124,999} Events are Mutually Exclusive. P(A or B) = P(A) + P(B) = 7/36 + 9/36 = 16/36 =4/9 =.444 (36 is the total number of months)

Decide whether the graph represents a discrete random variable or a continuous random variable. Explain your reasoning. The annual traffic fatalities in a country [line graph with six points]

Discrete, because number of fatalities is a random variable that is countable.

List an example of two events are dependent:

Drawing one card from a standard​ deck, not replacing​ it, and then selecting another card Not putting money in a parking meter and getting a parking ticket A father having hazel eyes and a daughter having hazel eyes

In a binomial​ experiement, what does it mean to say that each trial is independent of the other​ trials?

Each trial is independent of the other trials if the outcome of one trial does not affect the outcome of any of the other trials

Determine if mutually exclusive: Event A: Roll a 3 on a die Event B: Roll a 4 on a die

Event A has one outcome, a 3. Event B also has one outcome, a 4. These outcomes cannot occur at the same time, so the events are mutually exclusive

List an example of two events that are independent

Rolling a dice Tossing a coin and getting a​ head, and then rolling a​ six-sided die and obtaining a 6 Selecting a ball numbered 1 through 12 from a​ bin, replacing​ it, and then selecting a second numbered ball from the bin

Explain how the complement can be used to find the probability of getting at least one item of a particular type.

The complement of​ "at least​ one" is​ "none." So, the probability of getting at least one item is equal to 1 - P(none of the​ items).

A venn diagram shows one circle labeled PG movies and another not overlapping circle that says G movies. Are the events mutually exclusive?

The events are mutually exclusive, since there are no movies that are rated G and are rated PG.

Use the addition rule You roll a die. Find the probability of rolling a number less than 3 or rolling an odd number.

The events are not mutually exclusive because 1 is an outcome of both events. P (less than 3 or odd) = P(less than 3) + P(odd) - P(less than 3 and odd) = 2/6+3/6 - 1/6 =4/6 =2/3 =.667

Venn Diagram of Presidents Who won the State of California and an overlapping circle Presidents Who Won the Election. Are they mutually exclusive?

The events are not mutually exclusive, since there is at least 1 president candidate who won the State of California and lost the election.

What requirements are necessary for a normal probability distribution to be a standard normal probability​ distribution?

The mean and standard deviation have the values of mu equals 0 and sigma equals 1.

When you calculate the number of permutations of *n* distinct objects taken *r* at a​ time, what are you​ counting?

The number of ordered arrangements of n objects taken r at a time.

The problem involves a permutation because the order in which the letters are selected does matter

The outcome of a probability experiment is often a count or a measure. When this​ occurs, the outcome is called a random variable.

What is a random​ variable?

The outcome of a probability experiment is often a count or a measure. When this​ occurs, the outcome is called a random variable.

Conditional Probability

The probability of an event occurring, given that another event has already occured. The conditional probability of event B occurring, given that event A has occurred, is denoted by P(B | A) and is read: "Probability of B, given A"

The Addition Rule

The probablity that events A or B will occur, P(A or B) is given by: P(A or B) = P (A) + P (B) - P(A and B) If events A and B are mutually exclusive then: P(A or B) = P(A) + P (B)

How many different 7​-letter passwords can be formed from the letters Upper O​, Upper P​, Upper Q​, Upper R​, Upper S​, Upper T​, and Upper U if no repetition of letters is​ allowed?

The problem involves a permutation because the order in which the letters are selected does matter

Determine whether the random variable x is discrete or continuous. Explain. Let x represent the distance a baseball travels in the air after being hit.

The random variable is continuous​, because it has an uncountable number of possible outcomes.

Determine whether the random variable x is discrete or continuous. Explain. Let x represent the number of bald eagles in the country.

The random variable is discrete​, because it has a countable number of possible outcomes.

T/F A combination is an ordered arrangement of objects.

The statement is false. A true statement would be​ "A permutation is an ordered arrangement of​ objects."

Determine if the statement is true or false. If the statement is​ false, rewrite it as a true statement. The expected value of a random variable can never be negative.

The statement is false. The expected value of a random variable can be negative

Draw two normal curves that have the same mean but different standard deviations. Describe the similarities and differences.

The two curves will have the same line of symmetry. The curve with the larger standard deviation will be more spread out than the curve with the smaller standard deviation.

For the given pair of​ events, classify the two events as independent or dependent. Waking up and finding the alarm clock blinking 12 : 00 Getting to class late

The two events are dependent because the occurrence of one affects the probability of the occurrence of the other.

Determine whether the following events are mutually exclusive. Explain your reasoning. Event​ A: Randomly select a voter who is a registered Republican. Event​ B: Randomly select a voter who is a registered member of the Reform Party.

These events are mutually​ exclusive, since it is not possible for a voter to both be a registered Republican and be a registered member of the Reform Party.

Determine whether the following events are mutually exclusive. Explain your reasoning. Event​ A: Randomly select a female biology major. Event​ B: Randomly select a biology major who is 21 years old.

These events are not mutually​ exclusive, since it is possible to select a female biology major who is 21 years old.

Determine whether the statement is true or false. If it is​ false, rewrite it as a true statement. If two events are mutually​ exclusive, they have no outcomes in common.

True

T/F The number of different ordered arrangements of n distinct objects is​ n!

True

T/F When you divide the number of permutations of 11 objects taken 3 at a time by​ 3!, you will get the number of combinations of 11 objects taken 3 at a time.

True

*7*C*5* = *7*C*2*

True *n*C*r = n!/(n-r)!r! Therefore *n*C*r*=*n*C*n-r*. Notice that 7-5 =2

Mutally exclusive

Two events A and B are mutally exclusive when A and B cannot occur at the same time. That is, A and B have no outcomes in common. P ( A and B) = 0

independent

Two events are are independent when the occurence of one of the events does not affect the probability of the occurrence of the other event. Two events A and B are independent when: P (B | A) = P(B)

Find the mean variance and standard deviation of the binomial distribution when given values of n and p. n= 90 p=.4

mean = np 90 * .4 = 36 variance = npq q is the probability of failure in a single trail (q= 1- p) 90 * .4 * (1-.4) = 21.6 standard = sqrt(variance) sqrt(21.6) = 4.6

Why is it correct to say​ "a" normal distribution and​ "the" standard normal​ distribution?

​"The" standard normal distribution is used to describe one specific normal distribution left parenthesis mu equals 0 comma sigma equals 1 right parenthesis . ​"A" normal distribution is used to describe a normal distribution with any mean and standard deviation.

The probability that event A or event B will occur is P(A or B) - P(A) + P(B) - P(A or B)

​False, the probability that A or B will occur is P(A or B) = P(A) + P(B) - P(A and B)

In a normal​ distribution, which is​ greater, the mean or the​ median? Explain.

​Neither; in a normal​ distribution, the mean and median are equal.


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