Probability Terminology and Notation
A spinner is spun to determine the prize a game-show contestant wins. The spinner is divided into 10 equal sections. 4 of the sections are labeled $100, 5 of the sections are labeled $0, and the last section is labeled $500. If the contestant gets to spin the spinner 10 times, what is the amount of money the contestant should expect to w
$900
There are eight players on a basketball team. They are practicing their free throws by having each player shoot two free throws. The table below shows the results of each player's free throw attempts, where N represents a missed free throw and Y represents a made free throw. Construct the probability distribution of X for the number of free throws made by the players. Arrange x in increasing order and write the probabilities P(x) as simplified fractions.
X. 0 1. 2. P(X). 1/8. 1/2. 3/8 From the table, the players made either 0 throws, 1 throw, or 2 throws. So x can take 0, 1, or 2. For x=0, no throws are made in the two free throws by player 1. So, P(0)=18. For x=1, 1throw is made in the two free throws by player 3, player 4, player 6, and player 7. So, P(1)=48=12. For x=2, 2 throws are made in the two free throws by player 2, player 5 and player 8. So, P(2)=38. Thus,
Which of the following gives the definition of event? Select the correct answer below: the set of all possible outcomes of an experiment a subset of the set of all outcomes of an experiment a planned activity carried out under controlled conditions one specific execution of an experiment
a subset of the set of all outcomes of an experiment An event is defined as a subset of the set of all outcomes of an experiment.
otherwise you lose the $50. A standard roulette wheel has 38 slots numbered 00, 0, 1, 2, ... , 36. What is the expected profit for one spin of the roulette wheel with t
-$2.63
Find the expected value of the random variable X whose probability distribution is given below. x 1 2 3 4 p(X=x) 0.1 0.2 0.3 0.4
The expected value of a random variable X can be calculated as shown below. E(X)=∑ xp(x)=1(0.1)+2(0.2)+3(0.3)+4(0.4)=3
Which of the following best describes the term mutually exclusive? Select the correct answer below: all events which are not included in the specified event the property of two events in which the occurrence of one of the events affects the chance of the other occurring the property of two events in which the knowledge that one of the events occurred does not affect the chance the other occurs the property of events in which none can occur at the same time
the property of events in which none can occur at the same time Mutually exclusive is defined as the property of events in which none can occur at the same time.
Let C be the event that a randomly chosen cancer patient has received chemotherapy. Let E be the event that a randomly chosen cancer patient has received elective surgery. Identify the answer which expresses the following with correct notation: Of all the cancer patients that have received chemotherapy, the probability that a randomly chosen cancer patient has had elective surgery. Select the correct answer below: P(C|E) P(E|C) P(E) AND P(C) P(C AND E)
P(E|C) Remember that in general, P(A|B) is read as "The probability of A given B," or equivalently, as "Of all the times B occurs, the probability that A occurs also." So in this case, the phrase "Of all the cancer patients who have received chemotherapy" can be rephrased to mean "Given that a cancer patient has received chemotherapy," so the correct answer is P(E|C).
In a coin and die game, you roll a fair six-sided die and toss a coin. If you roll a 6 and toss a tails, you gain $110. Otherwise, you lose $10. If you were to play the game 45 times, how much money can you expect to gain or lose?
$0.00 Since there are 6 possible outcomes from rolling the dice and 2 possible outcomes from flipping the coin, there are 6×2=12 equally likely outcomes. Thus, rolling a 6 and tossing a tails is 1of the 12 possible outcomes of this game. Therefore, the probability of winning $110 is 112. The probability of getting any other outcome and losing the $10 is 1112. Multiply the outcome values by the probability to determine the expected gain or loss from one game.
In a game, you toss a fair coin and a fair six-sided die. If you toss a heads on the coin and roll either a 3 or a 6 on the die, you win $30. Otherwise, you lose $6. What is the expected profit of one round of this game?
$0.00 value=E1×P1+E2×P2 =$30×16+(−$6)×56 =$5.00−$5.00= $0.00
If we roll two dice, determine whether each of the following pairs of events are independent or dependent. (a) Event A is rolling a 1 on the first die. Event B is rolling a 2 on the second die. (b) Event A is rolling a 6 on the first die. Event B is getting a sum of more than 7 with the two dice.
(a) INDEPENDENT= Rolling a 1 on the first die does not affect the probability of rolling a 2 on the second die. So A and B are independent. (b) DEPENDENT= Imagine that event A has occurred. So we have rolled a 6 on the first die. If the second die is 2 or more, the sum of the dice will be greater than 7. That means that B has become more likely, so A and B are dependent.
Suppose you select a random card from a standard deck of 52 cards and no jokers. If you draw a face card (king, queen, or jack), you win $5. If you draw an ace, you win $20. If you draw any other card, you lose $2. What is the expected profit from any one game?
.$1.31 Out of the 52 playing cards in a standard deck, 12 of them are face cards. This corresponds to a probability of 1252, or 313. The probability of winning $5 by drawing one of these cards is 313. There are also 4 aces in a standard deck, so the probability of drawing an ace and winning $20 is 452, or 113. For the remaining 36 cards in the deck that you can draw, you lose $2. This corresponds to a probability of 3652, or 913. Multiply the outcome values by the probabilities to get the expected profit from one game
Which of the following pairs of events are independent? Select all correct answers. A) You roll a die and then flip a coin. Event A is getting a 5 or more with the die. Event B is getting heads with the coin. B) You roll a die and then flip a coin. Event A is getting a 4 or more with the die and getting tails with the coin. Event B is getting tails with the coin. C) You flip a coin twice. Event A is getting heads on the first flip. Event B is getting tails on the second flip. D) You flip a coin twice. Event A is getting heads on the first flip. Event B is getting heads on the first flip and heads on the second flip.
A) You roll a die and then flip a coin. Event A is getting a 5 or more with the die. Event B is getting heads with the coin. C) You flip a coin twice. Event A is getting heads on the first flip. Event B is getting tails on the second flip.
Which of the following pairs of events are independent? Select all correct answers. A) You roll a die twice. Event A is getting an even number on the first roll. Event B is getting a 4 on the second roll. B) You roll a die twice. Event A is getting a 6 on the first roll. Event B is getting a total of more than 7. C) You flip a coin and roll a die. Event A is getting heads on the coin. Event B is getting a 3 or more on the die. D) You roll a die and flip a coin. Event A is getting heads with the coin and getting 5 on the die. Event B is getting 3 or more on the die.
A) You roll a die twice. Event A is getting an even number on the first roll. Event B is getting a 4 on the second roll. C) You flip a coin and roll a die. Event A is getting heads on the coin. Event B is getting a 3 or more on the die. Answer A explained= What happens on the first roll does not affect what happens on the second roll. knowing event A happened does not affect the probability of event B. So they are independent. Answer C explained= What happens on the coin flip does not affect what happens on the roll of the die, the outcome of A does not affect the outcome of B. So A and B are independent.
suppose we flip a coin four times. Let event A be getting heads on the first three flips, and event B be getting all four heads. Events A and B are Independent or dependent?
Dependent events Note that B is fairly unlikely. But if we know that A has occurred, then there's a much better chance that B occurred too. So knowing about A affects the probability of B. This means A and B are dependent.
You flip a coin and then roll a die. Which of the following is an outcome of this experiment? Select all correct answers. Well done! You got it right. Heads on the coin 1 on the die Heads on the coin and 1 on the die Tails on the coin and 5 on the die
Heads on the coin and 1 on the die Tails on the coin and 5 on the die An outcome of the experiment includes the result of the coin flip and the result of the die roll. So "Heads on the coin and 1 on the die" and "Tails on the coin and 5 on the die" are both outcomes of the experiment.
In a die game, you roll a standard 6-sided die twice. If the second number rolled is the same as the first number rolled, you win $25. Otherwise, you lose $2. If you were to play the game 100 times, how much money can you expect to make?
If you play the game 100 times, you can expect to make $250.00 Out of the 36 possible combinations of two dice rolls, 6 of these consist of the same number twice. This corresponds to a probability of 636, or 16. The probability of earning $25 is 16. For the remaining 30 combinations, you lose $2. This corresponds to a probability of 3036, or 56. Multiply the outcome values by the probabilities to get the expected profit from one game. expected value=E1×P1+E2×P2=$25×16+(−$2)×56≈$4.17−$1.67=$2.50 This is the expected profit of one game. To get the expected profit of 100 games, multiply the expected value by 100. $2.50×100=$250.00
suppose we flip a coin twice. Let the event A be getting heads on the first flip and let the event B be getting tails on the second flip. Events A and B are Independent or dependent?
Independent events Note that the second coin flip is unaffected by what happened on the first flip. So whether A occurs or not, the probability of B is unchanged. Therefore, A and B are independent events.
Paul will roll two standard dice simultaneously. If Event A = both dice are odd and Event B = at least one die is even, which of the following best describes events A and B? Mutually Exclusive Not Mutually Exclusive Independent Dependent
Mutually Exclusive Dependent Mutually exclusive events are events that cannot occur at the same time. In this case, both dice cannot be odd (event A) if at least one of the dice is even (event B). Independent events are those for the occurrence of one event has no effect on the probability of the other, and dependent events are any that are not independent. Mutually exclusive events are almost always also dependent (with the exception of events that are already impossible) since the occurrence of one event means the probability of the other event changes 0.
In a die game, you roll a standard 6-sided die twice. If the second number rolled is the same as the first number rolled, you win $25. Otherwise, you lose $2. If you were to play the game 100 times, how much money can you expect to make?
Out of the 36 possible combinations of two dice rolls, 6 of these consist of the same number twice. This corresponds to a probability of 636, or 16. The probability of earning $25 is 16. For the remaining 30 combinations, you lose $2. This corresponds to a probability of 3036, or 56. Multiply the outcome values by the probabilities to get the expected profit from one game. expected value=E1×P1+E2×P2=$25×16+(−$2)×56≈$4.17−$1.67=$2.50 This is the expected profit of one game. To get the expected profit of 100 games, multiply the expected value by 100. $2.50×100=$250.00 If you play the game 100 times, you can expect to make $250.00
Explanation for these: Outcome: Event: Trial: Independent events: Dependent events: Mutually exclusive:
Outcome: any of the possible results of an experiment in probability. The Outcome aka Outcome space Event: some subset of the possible outcomes of an experiment, can be described by listing the outcomes or described in words Trial: one repetition or instance of a repeated experiment Independent events: events that have no influence on each other; if the fact that one event has occurred does not affect the probability that the other event will occur Dependent events: events that influence the occurrence of the other; if or not one event occurs does affect the probability that the other event will occur Mutually exclusive: events which are impossible to both occur or that have no outcomes in common. aka Disjoint events
Let B be the event that a randomly chosen person has low blood pressure. Let E be the event that a randomly chosen person exercises regularly. Identify the answer which expresses the following with correct notation: The probability that a randomly chosen person exercises regularly, given that the person has low blood pressure. P(E|B) P(B|E) P(E) AND P(B) P(B AND E)
P(E|B) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the person has low blood pressure, so the correct answer is P(E|B).
Let R be the event that a randomly chosen person has visited Rome, Italy. Let G be the event that a randomly chosen person has visited Greece. Place the correct event in each response box below to show: Given that the person has visited Rome, Italy, the probability that a randomly chosen person has visited Greece= P( ___|___ )
P(G | R ) Remember that in general, P(A|B) is read as "The probability of A given B". Here we want to know the probability that a person has visited Greece given that the person has visited Rome, Italy, so the correct answer is P(G|R).
Let S be the event that a randomly chosen store is having a sale. Let M be the event that a randomly chosen store has marked up their prices in the last six months. Identify the answer which expresses the following with correct notation: Given that the store is having a sale, the probability that a randomly chosen store has marked up their prices in the last six months. Select the correct answer below: P(M) AND P(S) P(M|S) P(S|M) P(S AND M)
P(M|S) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the store is having a sale, so the correct answer is P(M|S).
Let D be the event that a randomly chosen person has seen a dermatologist. Let S be the event that a randomly chosen person has had surgery for skin cancer. Identify the answer which expresses the following with correct notation: The probability that a randomly chosen person has had surgery for skin cancer, given that the person has seen a dermatologist. P(D|S) P(D AND S) P(S) AND P(D) P(S|D)
P(S|D) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the person has seen a dermatologist, so the correct answer is P(S|D).
Let G be the event that a randomly chosen employee of a restaurant is a General Manager. Let S be the event that a randomly chosen employee of a restaurant works at a seafood restaurant. Identify the answer which expresses the following with correct notation: Given that the employee is a General Manager, the probability that a randomly chosen employee of a restaurant works at a seafood restaurant. P(S|G) P(G|S) P(G AND S) P(S) AND P(G)
P(S|G) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the employee is a General Manager, so the correct answer is P(S|G).
Let M be the event that a randomly chosen student passes a math test. Let S be the event that a randomly chosen student studies every day. Identify the answer which expresses the following with correct notation: The probability that a randomly chosen student studies every day, given that the student passes a math test. Select the correct answer below: P(M|S) P(M AND S) P(S) AND P(M) P(S|M)
P(S|M) Remember that in general, P(A|B) is read as "The probability of A given B". Here we are given that the student passes a math test, so the correct answer is P(S|M).
Beth is performing an experiment to check if a die is fair. She rolls the die 5 times and records the sequence of numbers she gets. Which of these is an outcome of this experiment? Select all correct answers: Rolling a die Rolling a die five times Rolling the sequence 1,1,2,1,6 Rolling five 4's Rolling the sequence 1,1,2
rolling the sequence 1,1,2,1,6 Rolling five 4's An outcome is a specific result of an experiment. So the outcomes of this experiment are all the possible sequences of five die rolls. So in this case, a particular sequence such as 1,1,2,1,6 is an outcome. So is rolling five 4's, because this is a specific outcome (4,4,4,4,4).
Suppose you play a game where you toss three fair coins. If you get three tails, you win $10. Otherwise, you lose $2. If you were to play this game 15 times, how much would you expect to gain or lose?
If you play the game 15 times, you can expect to lose $7.50.
A business firm has 800 employees out of which 176 are married men, 272 are married women, 208 are single men, and 144 are single women. Let x=0, 1, 2 and 3 represent the events that a person is a married man, married woman, single man and single woman, respectively. Find the probability distribution. Arrange x in increasing order and write the probabilities P(x) as simplified fractions. X __ __ __ __ p(x) __ __ __ __
X 0 1 2 3 P(x) 176/800 272/800 208/800 144/800
If we roll two dice, which of the following pairs of events are mutually exclusive? (a) Event A is rolling a 1 on the first die. Event B is rolling a 2 on the second die. (b) Event A is rolling a 6 on the first die. Event B is getting a sum of less than 7 with the two dice. (c) Event A is rolling a 3 on the first die. Event B is getting a sum of 9 with the two dice.
(b) Event A is rolling a 6 on the first die. Event B is getting a sum of less than 7 with the two dice. Imagine that event A has occurred. So we have rolled a 6 on the first die. No matter what we roll on the second die, it will be at least 1, so the sum of the two dice will be at least 7. Therefore, there is no chance that B can occur. Thus, we see that it is impossible for A and B to both occur, so A and B are mutually exclusive.
In a card game, there are 15 cards laid out on a table. 6 of the cards are blank, 2 of the cards are labeled $3, and the remaining cards are labeled $1. When you select a card at random, you earn what is labeled on the card. After each game, you return the card and the cards are shuffled. Suppose you play this game 10 times. How much mon
From 10 games, you can expect to gain $8.67
Let R be the event that a randomly chosen person has red hair. Let G be the event that a randomly chosen person has green eyes. Place the correct event in each response box below to show: Given that the person has red hair, the probability that a randomly chosen person has green eyes. P( __|__ )
P(G | R ) Remember that in general, P(A|B) is read as "The probability of A given B". Here we want to know the probability that a person has green eyes given that the person has red hair, so the correct answer is P(G|R).
Using a standard 52-card deck, Michelle will draw 6 cards with replacement. If Event A = drawing all hearts and Event B = drawing no face cards, which of the following best describes events A and B? independent dependent mutually exclusive complement
independent Events are independent if the knowledge of one event occurring does not affect the chance of the other event occurring. In this case, the chance of either event occurring is the same whether or not the other event occurs.
Arianna will roll a standard die 10 times in which she will record the value of each roll. What is a trial of this experiment? Select the correct answer below: one roll of the die rolling at least one 5 ten rolls of the die rolling a sum of 40
one roll of the die A trial is one specific execution of an experiment. In this case, each trial is one roll of the die.
In a die game, you roll a fair six-sided die. If you roll a 2, you win $25. If you roll anything else, you lose $5. What is the expected profit of one roll in this game? Round your answer to the nearest cent. Enter an expected loss as a negative number.
$0.00 Rolling a 2 is 1 out of 6 possible outcomes, which is a probability of 16. In our game, the probability of winning $25 is 16. For the remaining numbers we can roll, we lose $5. This has a probability of 56. Multiplying the events by the outcomes, we get the expected profit for each time the game is played. expected value= E1×P1+E2×P2 =$25×16+(−$5)×56 ≈$4.17−$4.17 =$0.00
You toss a coin three times. If you toss heads exactly two times, you win $2. If you toss heads all three times, you win $8. Otherwise, you lose $3. What is the expected payout for one round of this game?
$0.25
An automotive insurance company is reviewing a customer's application for a one-year policy. Based on the customer's driving history and the insurance company's past experience, the company assumes that the probability of each payout for one year is as shown in the table provided. What is the expected payout for the insurance company?
$1335
An insurance company sells a homeowner's insurance policy that is $600 per year per policy. The insurance company predicts that the probability of having a home being damaged in the area it sells this policy is about 11,000. If the home is damaged, the insurance company expects to pay $400,000 to the homeowner. If the company were to sell 250 policies, how much money can it expect to gain or lose?
$50,000
A flood insurance company sells policies for $700 per year. If a customer's house is flooded, they are given $250,000 for repairs. The insurance company has calculated the chances that a house is flooded to be 12,500 over the year. How much money can the insurance company expect to make with each policy sold?
$600
Patricia will draw 8 cards from a standard 52-card deck with replacement. Which of the following are NOT events in this experiment? drawing 8 hearts drawing 8 diamonds drawing 1 card drawing 4 aces and 4 kings
drawing 1 card Events are any combinations of outcomes or particular results in an experiment. Drawing 8 cards is the experiment and drawing 1 card is a trial of the experiment, neither of which specify a result or outcome.
In a day care center each child takes three classes and either passes or fails each class. The grades are shown in the table below. P represents a passed class, and F represents a failed class. Let X represent the number of classes failed by a child. Construct a probability distribution for X. Arrange x in increasing order and write the probabilities P(x) as simplified fractions. Child Grades Child 1 PFP Child 2 PPF Child 3 FFP Child 4 PPP Child 5 PFP Child 6 PPF Child 7 FPF Child 8 PPP
x 0 1 2 p(x) 1/4 1/2 1/4 Here X represents the number of classes failed by a child. In the given table, the number of classes a child fails is either 0, 1, or 2. So, x takes 0, 1, or 2. If x=0, then the child passes all the classes. In the given table, child 4 and child 8 pass all the classes. So, P(0)=28=14. If x=1, then the child fails exactly one class. In the given table, child 1, child 2, child 5, and child 6 fail exactly one class. So, P(1)=48=12. If x=2, then the child fails exactly two classes. In the given table, child 3 and child 7 fail two classes. So, P(2)=28=14. Thus, the probability distribution is shown below.
An altered dice has one dot on one face, two dots on three faces, and three dots on two faces. The dice is to be tossed once. Let X be the number of dots on the upturned face. Construct a table showing the probability distribution of X . Arrange x in increasing order and write the probabilities P(x) as simplified fractions.
x. 1. 2. 3. P(x) 1/6. 3/6. 2/6
