Test 2
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.
