QMST Test Two
A standard pair of six-sided dice is rolled. What is the probability of rolling a sum less than 3?
1/36
A person tosses a coin 6 times. In how many ways can he get 4 tails?
6 nCr 4 = 15
A variable whose numeric value is determined by the outcome of a random experiment is called a...
Random Variable
Determine whether each of the following probabilities is subjective, empirical, or classical. The probability that G32 will be called first in a bingo game.
Assuming that there are 75 numbers in a standard bingo game, all 75 outcomes are known, so this is an example of a classical probability.
Determine whether the following events are mutually exclusive. Choosing a seven or a spade out of a standard deck of cards.
Not Mutually Exclusive
Consider drawing a card from a standard 52-card deck. What is the probability of drawing a spade?
There are 13 spades in the deck, so the probability of choosing a spade is: P(spade) = n(E)/n(S) = 13/52 = 1/4 = 0.25
The Empirical Probability, or the probability that E, the event occurs, is P(E)=
number of times that event E occurs (f) / total number of times the expirment is performed (n)
Three cards are drawn with replacement from a standard deck. What is the probability that the first card will be a diamond, the second card will be a red card, and the third card will be an ace?
(13/52) * (25/52) * (4/52) = 1/104
****End of Chapter 4.2b -Probability Rules: Independence, Multiplication Rules, and Conditional Probability****
****End of Chapter 4.2b - Probability Rules: Independence, Multiplication Rules, and Conditional Probability****
***Start of Chapter 4.2a - Probability Rules: Properties, the Complement, and Addition Rules***
***Start of Chapter 4.2a - Probability Rules: Properties, the Complement, and Addition Rules***
A box contains 6 red marbles, 34 white marbles, and 64 blue marbles. If a marble is randomly selected from the box, what is the probability that it is white? Express your answer as a simplified fraction or a decimal rounded to four decimal places.
0.3269
A coin is tossed 6 times. What is the probability of getting all heads? Express your answer as a simplified fraction or a decimal rounded to four decimal places.
1/2^6 = 0.0156
There are 16 people in an office with 6 different phone lines. If all the lines begin to ring at once, how many groups of 6 people can answer these lines?
Choose the person to answer the first phone in 16 ways. That leaves 15 people to choose to answer the second phone. That leaves 14 people to choose to answer the third phone and so on Answer 16 * 15* 14 * 13 * 12 * 11 = 16 nCr 6 = 8008 ways
Calculate the following factorial expressions. (8!)/(4!(6-2)!)
Make sure that you begin by subtracting 6−2 70
Calculate the following factorial expressions. 6!
Multiply each positive integer less than or equal to 6. 6! = (6)(5)(4)(3)(2)(1) = 720
Determine whether the following value could be a probability. 32/21
No A probability must be between zero and one.
This is a model which describes a specific kind of random process. It can be a table or formula that lists the probabilities for each outcome of the random variable, X.
Probability Distribution
Determine whether each of the following probabilities is subjective, empirical, or classical. A teacher predicts that 20% of his class will get A's on the final.
The teacher is giving an educated guess, thus a subjective probability.
Two events are ______________________ if one event happening does not influence the probability of the other event happening.
independent
If the probability of an event is __________, then it cannot occur.
zero
Write out the sample space for the given experiment. Use the following letters to indicate each choice: W for white, Y for yellow, B for blue, C for cherry, E for ebony, and T for teak. While renovating your house, you have a choice of paint colors for your game room: white, yellow, or blue. You also have the following options for the finish on your entertainment center: cherry, ebony, or teak.
{ WC, WE, WT, YC, YE, YT, BC, BE, BT }
States that you can multiply together the number of possible outcomes for each stage in an experiment in order to obtain the total number of outcomes for that experiment.
Fundamental Counting Principle
Calculate the following factorial expressions. 67!/65!
It would be very cumbersome to multiply out 67! and 65! and then divide. Instead, we will cancel first, so 67 * 66 = 4422
Determine whether the following events are mutually exclusive. Choosing a student who is a French major or a chemistry major from a nearby university to participate in a research study. (Assume that each student only has one major.)
Mutually Exclusive
***End of Chapter 4.2a - Probability Rules: Properties, the Complement, and Addition Rules***
***End of Chapter 4.2a - Probability Rules: Properties, the Complement, and Addition Rules***
The newly elected president needs to decide the remaining 5 spots available in the cabinet. If there are 14 eligible candidates for these positions (where rank matters), how many different ways can the members of the cabinet be appointed?
14 nPr 5 = 240,240
4 cards are drawn from a standard deck of 52 playing cards. How many different 4-card hands are possible if the drawing is done without replacement?
270725
3 cards are drawn from a standard deck of 52 playing cards. How many different 3-card hands are possible if the drawing is done without replacement?
52C3 = 22100
Out of 450 applicants for a job, 212 are female and 55 are female and have a graduate degree. What is the probability that a randomly chosen applicant has a graduate degree, given that they are female? Express your answer as a fraction or a decimal rounded to four decimal places.
55/212 or 0.2594
Danielle's dog just had 8 puppies. 3 of the puppies are entirely black and the other 5 are black with white spots. If Danielle picks up a puppy at random, what is the probability that it will have spots?
The problem says that there are 8 puppies in total and that 5 of the puppies have spots, so the probability is: P(spotted puppy) = n(E)/n(S) = 5/8 = 0.625
Consider drawing a card from a standard 52-card deck. What is the probability of drawing a red card?
There are two red suits (diamonds and hearts) and each suit has 13 cards, so there are 26 red cards in the deck. The probability of choosing a red card is then: P(red card) = n(E)/n(S) = 26/52 = 1/2 = 0.5
In three tosses of a fair coin, what is the probability that two of the three tosses will come up heads?
Using H for heads and T for tails, the sample space for the experiment with equally likely outcomes can be written as the set of 8 outcomes. S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Here the notation HTT, for instance, represents the possible outcome of heads on the first toss and tails on the remaining two tosses. If E is the event of getting "two heads" in three tosses, then E can be written as the set of 3 outcomes E = {HHT, HTH, THH}. Then, applying the classical probability formula, we have: P(two heads) = P(E) = n(E)/n(S) = 3/8
When the order in which the objects are chosen is not important, then we use a ____________________.
combination
Consider the sample space for a probability experiment, and one event, E, in that sample space. The ____________________ for E, denoted Ec, consists of all outcomes in the sample space that are not in E.
complement
When two events are not independent, the outcome of one influences the probability of the other, this is called ___________________________.
conditional probability
If the probability of an event is __________, then it will absolutely occur.
one
Complete the given sentence. Each individual result of a probability experiment is called a(n)
outcome
This most precise type of probability, ______________________ is calculated by taking all possible outcomes for an experiment into account.
Classical Probability (also known as Theoretical Probability)
A standard pair of six-sided dice is rolled. What is the probability of rolling a sum greater than 7? Express your answer as a fraction or a decimal number rounded to four decimal places.
5/12 or 0.4167
How many different ways can the letters in the name Smith be arranged to form five letter "words"?
5⋅4⋅3⋅2⋅1 = 120
Decide if the following probability is classical, empirical, or subjective. Selecting a 7 from a standard deck of cards.
Classical Probability
A random variable which has a countable number of possible outcomes.
Discrete Random Variable
Determine if the following events are independent. Robert makes it to work on time. Kathy wins the state lottery.
Independent
Determine whether the following events are mutually exclusive. Choosing a heart or a black card out of a standard deck of cards.
Mutually Exclusive
Calculate the following factorial expressions. (8!)/((4-2)!)
You must begin by subtracting the expression in parentheses. Next, calculate each factorial expression and then divide after canceling out, leaving us only with; 8 * 7 * 6 * 5 * 4 * 3 = 20,160
The ____________________ of a random variable should be very close to the average value of a large number of observations from the random process, and the larger the number of observations collected, the more likely the average of the observations will be close to the expected value.
expected value
For a ____________________, the order in which the objects are chosen is important.
permutation
Three cards are drawn with replacement from a standard deck. What is the probability that the first card will be a spade, the second card will be a red card, and the third card will be the five of clubs?
(13/52) * (26/52) * (1/52) = 1/416
How many ways can Carlos choose 2 pizza toppings from a menu of 4 toppings if each topping can only be chosen once?
(4 * 3)/(2 * 1)
A box contains 13 green marbles and 13 white marbles. If the first marble chosen was a white marble, what is the probability of choosing, without replacement, another white marble? Express your answer as a fraction or a decimal number rounded to four decimal places.
12/25 or 0.48
You are ordering a new home theater system that consists of a TV, surround sound system, and DVD player. You can choose from 16 different TVs, 23 types of surround sound systems, and 11 types of DVD players. How many different home theater systems can you build?
16 * 23 * 11 = 4048
A credit card company classifies its customers by gender and location of residence. The research department has gathered data from a random sample of 1673 customers. The data is summarized in the table below. Apartment: 187 males, 217 females Dorm: 101 males, 71 females With parents: 63 males, 229 females Frat/Sorority: 110 males, 222 females Other: 261 males, 212 females What is the probability that a customer is female and lives in 'Other'?
212/1673 = 0.1267
Find the number of outcomes in the complement of the given event. Out of 318 shoes in a department store, 182 are women's shoes.
318-182 = 136
An eight-sided die, which may or may not be a fair die, has four colors on it; you have been tossing the die for an hour and have recorded the color rolled for each toss. What is the probability you will roll an orange on your next toss of the die? Express your answer as a simplified fraction or a decimal rounded to four decimal places. Purple - 40 Green - 39 Brown - 29 Orange - 30
0.2174 or 5/23
Three cards are drawn with replacement from a standard deck. What is the probability that the first card will be a heart, the second card will be a black card, and the third card will be a face card?
3/104 or 0.0288
You are ordering a hamburger and can get up to 8 toppings, but each topping can only be used once. You tell the cashier to surprise you with the toppings you get. What is the probability that you get 1 topping? Express your answer as a fraction or a decimal number rounded to four decimal places.
8 nCr 1 = 70 (1+1)^8 = 256 70/256 = 1/32
Describe the complement for each of the following events. Choose a face card out of a standard deck of cards.
A standard deck of 52 cards has 12 face cards. If all 12 face cards are contained in the event, then the complement contains all 40 non-face cards.
Choose two cards from a standard deck, with replacement. What is the probability of choosing a two and then a five?
Because the cards are replaced after each draw, the two events are independent. Using the Multiplication Rule for Probability, we have. (4/52) * (4/52) = 1/169 or 0.0059
How many different car license plates are possible if a license plate consists of three numbers followed by three letters of the alphabet?
Each of the first 3 slots has 10 different choices (the digits 0-9). Each of the last 3 slots has 26 different choices (the 26 letters in the alphabet). Thus, by the Fundamental Counting Principle the total number of plates possible is 10⋅10⋅10⋅26⋅26⋅26 = 17,576,000.
Describe the complement for each of the following events. Out of 45 students in your history class, 25 have digital cable.
If 25 students have digital cable, then the complement contains the other 20 students who do not have digital cable (aka losers).
An educated guess regarding the chance that an event will occur. The accuracy of the probability depends on the expertise of the person giving the probability.
Subjective Probability
How many different ways can you arrange the letters in the word Mississippi?
We can make no distinction between each "i", "s", and "p" in the word, so we need to group the letters together. The letters of Mississippi are grouped as follows: k1=the number of M's=1 k2= the number of I's =4 k3= the number of S's=4 k4= the number of P's=2 Notice that k1+k2+k3+k4=1+4+4+2=11=n. The number of ways to arrange the letters in Mississippi is 34,650
How do we find the probability of students that are not male or do not live in a dorm?
We just exclude the male dorm dwellers.
An ____________________ is a subset of outcomes of the sample space.
event
In a given probability experiment, each individual result that is possible is called an ____________________.
outcome
A coordinator will select 4 songs from a list of 14 songs to compose an event's musical entertainment lineup. How many different lineups are possible?
Apparently order matters in this situation, so: 14 nPr 4 = 24024
Suppose you like to keep a jar of change on your desk. Currently, the jar contains the following: 11 Pennies 13 Nickels 26 Dimes 5 Quarters What is the probability that you reach into the jar and randomly grab a nickel and then, without replacement, a dime?
(13/55) * (26/54) = 169/1485 or 0.1138
****Start of Chapter 4.2b -Probability Rules: Independence, Multiplication Rules, and Conditional Probability****
****Start of Chapter 4.2b -Probability Rules: Independence, Multiplication Rules, and Conditional Probability****
****Start of Chapter 4.3 -Counting Rules****
****Start of Chapter 4.3 -Counting Rules****
****Start of Chapter 5.1 - Discrete Random Variables****
****Start of Chapter 5.1 - Discrete Random Variables****
***End of Chapter 4.1 - Classical Probability***
***End of Chapter 4.1 - Classical Probability***
What is the probability that a randomly selected person will have a birthday in April? Assume that this person was not born in a leap year. Express your answer as a simplified fraction or a decimal rounded to four decimal places.
3 days in April divided by 365 days in a year = 6/73 or 0.0822
Out of 450 applicants for a job, 212 are female and 55 are female and have a graduate degree. If 93 of the applicants have graduate degrees, what is the probability that a randomly chosen applicant is female, given that the applicant has a graduate degree? Express your answer as a fraction or a decimal rounded to four decimal places.
55/93 or 0.5914
A random variable that can assume any value on a continuous segment(s) of the real number line.
Continuous Random Variable
A ____________________ is the product of all positive integers less than or equal to n.
factorial
How many ways can four people be chosen from a group of twenty to serve on a committee?
Since there is no ranking in the committee order does not matter. Therefore, this is a combination problem. The sample size is 20, so n=20. Since four people are to be chosen, r=4. 20C4 = 20!/(4!(20-4)!) = 20!/(4!16!) = 4845
Decide if the following probability is classical, empirical, or subjective. You believe that you have a 4% chance of getting into an accident on your way to work.
Subjective Probability
Roll a pair of dice. What is the probability that neither die is a 5?
We could list every combination of the dice that does not have a five, but that would be tedious. It is much easier to count the outcomes in the complement. The complement of this event contains the outcomes in which either die is a five. (Check yourself by making sure that adding the event and its complement cover the entire sample space.) Let's list these outcomes. There are 11 outcomes where one of the dice is a 5 and a total sample space of 36 outcomes. Then, P(Ec) = 11/36. Subtracting this probability from 1 gives us P(E) = 1 − 11/36 = 25/36 ≈ 0.6944
A ____________________, or trial, is any process in which the result is random in nature, such as flipping a coin, tossing a pair of dice, or drawing a raffle ticket. In each case, there is more than one possible result and the result is determined at random.
probability experiment
For example, let's pretend that the experiment is rolling a 6-sided die. If E is the event of rolling an even number (2,4,6), then its complement is...
rolling an odd number (1,3,5).
The set of all possible outcomes for a given probability experiment is called the ____________________.
sample space
Write out the sample space for the given experiment. Use the letter R to indicate red, G to indicate green, and B to indicate blue. A die shows 3 different colors on it. Give the sample space for the next 2 rolls.
{RR,RG,RB,GG,GR,GB,BR,BG,BB}
First, a probability is always...
a number between 0 and 1. That is, 0≤P(E)≤1.
if you are choosing numbers with repetition for a bank pin number then...
you are allowed to repeat numbers, such as in the pin 1231,
***Start of Chapter 4.1 - Classical Probability***
***Start of Chapter 4.1 - Classical Probability***
You need to have a password with 4 letters followed by 3 even digits between 0 and 9, inclusive. If the characters and digits cannot be used more than once, how many choices do you have for your password?
26 * 25 * 24 * 23 * 5 * 4 * 3 = 21,528,000
A value meal package at Ron's Subs consists of a drink, a sandwich, and a bag of chips. There are 3 types of drinks to choose from, 4 types of sandwiches, and 4 types of chips. How many different value meal packages are possible?
3 * 4 * 4 = 48
You are getting a line-up ready for a school kickball game. You have 7 girls and 7 boys. The rules state each child must kick the same number of times and alternate girl-boy or boy-girl. How many ways can a line-up be made for one round of kicking?
7! * 7! * 2 = 50803200
What is the probability of choosing a black card from the deck, given that the first card was a spade? Assume the cards are chosen without replacement.
Because the first card was a spade, there are only 25 black cards left in the deck instead of 26. There are also only 51 cards total left in the deck. Thus, the conditional probability is 25/51 = 0.4902
Decide if the following probability is classical, empirical, or subjective. You calculate that the probability of randomly choosing a student who is from out-of-state is about 34%.
Empirical Probability
For a committee of 8 people, how many ways can a chairperson and a secretary be selected from amongst its members?
First note that the order of the members chosen is important, i.e., it is different if someone is elected for chairperson rather than secretary. Therefore, this is a permutation. The sample size is 8, so n=8. Since two positions are to be filled, r=2. 8P2 = 8!/(8-2)! = 56
Suppose a restaurant offers a value meal consisting of a sandwich and a drink. If there are 5 types of sandwiches and 4 types of drinks, how many different meals are possible?
In this problem there are two tasks to be performed, namely, choosing a sandwich and choosing a drink. For each sandwich choice, there are 4 possibilities for the drink choice. 5 * 4 = 20 possible meals or outcomes.
Find the probability of choosing either a heart or a face card (king, queen, jack) out of a standard deck of cards.
The key word here is "or". Using the Addition Rule for Probability, we have: P(heart or face card) P(heart)+P(face card)− P(heart and face card) = 13/52 + 12/52 − 3/52 = 22/52 = 11/26 or 0.4231.
Determine whether each of the following probabilities is subjective, empirical, or classical. An optometrist wants to know the probability that children will need glasses by the end of 5th grade. She finds that the number of students at the local elementary school (with grades K-5) is 432 and that 108 of them have glasses. From this she determines that the probability is 0.25.
The optometrist's probability is based on the statistics from the local elementary school. Since all children were not included, the probability is empirical.
Timothy has 15 books on his shelf: 5 mysteries, 3 suspense, 4 science fiction, 3 history. If he chooses a book at random off of his shelf, what is the probability that it will be a mystery?
The problem says that Timothy has 15 books total and there are 5 mysteries, so the probability is: P(mystery) = n(E)/n(S) = 5/15 = 1/3 ≈ 0.3333
What is the probability of choosing two non-face cards in a row? Assume that the cards are chosen without replacement.
We are dealing with dependent events, so we must use the Multiplication Rule for Dependent Events. When the first card is picked, all 40 non-face cards are available out of 52 cards. When the second card is drawn, there are only 39 non-face cards left out of 51 cards left in the deck. Thus we have: (40/52) * (39/51) = 1560/2652 or 0.5882
What is the probability of rolling a sum of 44 on a standard pair of six-sided dice? Express your answer as a fraction or a decimal number rounded to three decimal places, if necessary.
ince for every possible roll on one of the dice there are six possible rolls on the other, there are 36 possible rolls (combinations) of the pair. Now, there are 3 possible ways of getting a sum of 4 for the pair: (1)a pair of twos, (2) a 1 on one of the dice and a 3 on the other, or (3) vice versa of (2). So 3 out of 36 possible rolls yields 3/36 which reduces to 1/12.
When two events are independent, the probability that both events occur can easily be calculated by...
multiplying their respective probabilities together. P (E and F) = P(E)*P(F).
The Law of Large Numbers says that the greater the number of trials...
the closer the empirical probability will become to the true probability.
When you put an event and its complement together, you will always have...
the entire sample space.
Write out the sample space for the given experiment. Use the following letters to indicate each choice: M for mushrooms, O for olives, T for turkey, B for bacon, R for ranch, and V for vinaigrette. When deciding what you want to put into a salad for dinner at a restaurant, you will choose one of the following extra toppings: mushrooms, olives. Also, you will add one of following meats: turkey, bacon. Lastly, you will decide on one of the following dressings: ranch, vinaigrette.
{MTR,MTV,MBR,MBV,OTR,OTV,OBR,OBV}
****End of Chapter 4.3 - Counting Rules****
****End of Chapter 4.3 - Counting Rules****
A box contains 19 large marbles and 11 small marbles. Each marble is either green or white. 6 of the large marbles are green, and 5 of the small marbles are white. If a marble is randomly selected from the box, what is the probability that it is large or white? Express your answer as a fraction or a decimal number rounded to four decimal places.
19 large marbles + 5 small white marbles = 24 marbles that are large or white. 24/30 = 4/5 or 0.8
Suppose that a track coach is putting a 4×100 relay team together. How many different ways can she order those 4 runners?
The order is important, so this is a permutation problem. The sample size is 4, so n=4. However, for this problem r is also 4 because all 4 runners will be chosen to run a leg of the race. 4P4 = 4!/(4-4)! = 24