STATISTICS MODULE 7- PROBABILITY QUESTIONS WITH ANSWERS 2024-2025 UPDATED LATEST GRADED 10, Exams of Probability and Statistics

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STATISTICS MODULE 7- PROBABILITY

QUESTIONS WITH ANSWERS 2024-

UPDATED LATEST GRADED 100%

PASS.

Examples of Quantitative and Qualitative Probability - Answ Qualitative Description= impossible Quantitative Probability = 0% probability Qualitative Description= Unlikely Quantitative Probability = 1% to 30% probability Qualitative Description= As likely as unlikely Quantitative Probability = 40-60% Qualitative Description= likely Quantitative Probability = 70-99% Qualitative Description= certain Quantitative Probability = 100% experiment - Answ the procedure or situation in which the probability is being calculated example- rolling a dice, raining outcomes - Answ the possibilities of what can occur during the experiment, the results rolling a dice- 1,2,3,4,5,6 are outcomes event - Answ comprised of 1 or more outcomes die landing on an uneven number is an event probability of an event - Answ P(E)= probability of a certain event occurring

if R= RAIN and the weatherman says there is a 40% chance of rain, then P(R)=. an experiment is fair if - Answ each outcome is equally likely An experiment is the procedure to test the occurrence of an event. True or False? - Answ False Correct. This is a false statement. An experiment is the procedure for which the probability of an event is calculated. set - Answ simply a collection of unique elements. For example, a set of tree species is: oak, juniper, elm, maple. subsets - Answ Set A is a subset* of set B , if every element in A is contained within B. For example: A={1,2,3} B={1,2,3,4,5} A is a subset of B , because every element in set A is contained within set B. empty set - Answ has no elements {} For example, let's say you wanted to list the days of the week that do not end in a y. There are none! Therefore, this is the empty set. In set notation, the empty set is written as a pair of brackets with nothing between them: union - Answ The union* of two sets is a collection of all of the elements listed in the sets. For example: C={2,4,6} D={1,3,5} The union of C and D is {1,2,3,4,5,6} , as those are all of the elements that appear in the sets.

intersection - Answ The intersection* of two sets is a collection of the elements listed in both of the sets. For example: E={0,10,100} F={−2,−1,0,1,2} The intersection of E and F is {0} , as 0 is the only element that appears in both sets. theoretically probability - Answ is calculated as the number of ways one particular event can occur in a random experiment, divided by the total number of possible outcomes: if bob has 3 blue suits, 1 red and 1 orange what is the probability he will randomly pick a blue suit? 3/ Empirical (Observational) Probability - Answ gathers data by performing multiple experiments, or trials, and recording the results each time. For example, to have a better idea of the probability that a team will win a particular game, we naturally examine what other games it has won and against which teams. relative frequency - Answ how often the event occurs in the series of trials (or experiments) relative to the number of trials. a way to approximate a percentage by dividing the number of times an event occured in an experiment by the total number of trials this is influenced by random experiments random experiments - Answ are trials in which the outcome is not known ahead of time and the result does not depend on the results of other trials. For example, you can flip a coin ten times, and each will be a random experiment. You do not know ahead of time whether you will get heads or tails. Getting heads or tails on one flip does not help you predict the result of the next flip.

Relationship between Theoretical and Empirical Probabilities - Answ We can discover the relationship between the theoretical and empirical probabilities by running experiments for which the theoretical probability is accurate, such as flipping a coin. The probability that someone will flip a "heads" in a coin toss is the number of desired outcomes ( 1 , the head side) divided by the total number of possibilities ( 2 , heads or tails). 1/2 is 0.5 or 50% law of large numbers - Answ states that as the number of trials increases, the relative frequency of an event will converge on the theoretical probability. Simply put, the more times you flip a coin, the closer the relative frequency of heads will be to 50%. Joannie chooses a dessert recipe to bake for her book club meeting from the 25 dessert recipes she has in her recipe box. To predict the likelihood of her making a chocolate chip cookie recipe next book club meeting, would the theoretical probability 1/25 be most accurate? - Answ No Correct. The answer is b. She likely prefers some recipes over others or more often has the ingredients at hand. To help predict the future, an empirical probability will work best. A die is rolled 10,000 times. We would most likely expect the relative frequency of rolling a " 1 " to converge on the value 1/6. True or False? - Answ True Correct. This is a true statement. 10,000 tosses is a large number. By the law of large numbers we would expect the relative frequency to be close to 16, the theoretical probability. After a certain number of trials the empirical probability of an outcome will equal the theoretical probability. True or False? - Answ False Correct. This is a false statement. The empirical probability will always be an estimate. A researcher is testing the hypothesis that more screen time decreases a person's ability to read social clues. The researcher is using surveys to gather information both about a person's screen time and about his or her ability to read social clues in order to be able to predict how well a person

with a certain amount of screen time will be able to read social clues. How many surveys should the researcher collect? - Answ As many as possible. The law of large numbers holds that the more trials, the closer the empirical data comes to estimating the true probability. Correct. The answer is d. There is no way to calculate the theoretical probability in a case like this. The more data the researcher gathers the clearer an idea he or she will have about the extent of its influence (if any.) You examine the number of times a player hit a home run this season, divided by the number of at-bats the player had. What is this an example of? - Answ b) Empirical Probability Correct. The answer is b. We are using the number of times an event occurred to estimate the probability, therefore this is an example of empirical probability. sample space - Answ The sample space of flipping a coin is heads and tails. The sample space of rolling a regular six-sided die is 1 , 2 , 3 , 4 , 5 , and 6. sample size - Answ the number of different outcomes. We need the sample space to determine the sample size, and we need the sample size to calculate probability. The sample size of flipping a coin is 2 ; the sample size of rolling a six-sided die is 6. a list - Answ One very straightforward way to determine a sample space is to write out all the options in a list. The downside of this method is it can be difficult to determine if you have missed an outcome. To show the method, suppose someone who is planning to have two children wants to know the probability of having one boy and one girl. To calculate the probability, you first must determine the sample space of possible outcomes. This is a relatively easy sample space to find. Sample Space of Having 2 Children Boy Boy Boy Girl Girl Boy Girl Girl

You can see from the table above that the sample size is 4 and there are two ways to have a boy and a girl. The probability of having 1 boy and 1 girl when having 2 children is 24=. When making a list, be sure to create as much of a pattern as you can to ensure you have exhausted all possibilities. For example, you might organize the list above by starting with the maximum number of boys you can get. A 20 year study creates different categories of 40 year old patients: those with a greater inherited risk of heart disease and those without; those who have healthy eating habits at the start of the study and those who do not; those who get heart disease before age 50 , after the age of 50 but before age 60 , and those who do not yet have heart disease at the end of the study. - Answ List: inherited risk-healthy eating-heart disease before age 50 inherited risk-healthy eating-heart disease between 50 — 60 inherited risk-healthy eating-no heart disease by 60 inherited risk-unhealthy eating-heart disease before 50 inherited risk-unhealthy eating-heart disease between 50 — 60 inherited risk-unhealthy eating-no heart disease by 60 no inherited risk-healthy eating-heart disease before 50 no inherited risk-healthy eating-heart disease between 50 — 60 no inherited risk-healthy eating-no heart disease by 60 no inherited risk-unhealthy eating-heart disease before 50 no inherited risk-unhealthy eating-heart disease between 50 — 60 no inherited risk-unhealthy eating-no heart disease by 60 There are 12 different categories in the study What is the size of the sample space for flipping 3 coins? - Answ From the list constructed below, there are 8 outcomes. Heads, Heads, Heads Heads, Heads, Tails Heads, Tails, Heads Heads, Tails, Tails Tails, Tails, Tails Tails, Tails, Heads Tails, Heads, Tails Tails, Heads, Heads

What is the size of the sample space for rolling 1 die and flipping 1 coin? - Answ 1 , Heads 1, Tails 2, Heads 2, Tails 3, Heads 3, Tails 4, Heads 4, Tails 5, Heads 5, Tails 6, Heads 6, Tails 12 What is the size of the sample space for the order in which marbles are drawn out of a bag that contains 1 red, 1 yellow, and 1 green marble? - Answ The answer is 6. From the list that was constructed, there are 6 outcomes. Red, Yellow, Green Red, Green, Yellow Yellow, Red, Green Yellow, Green, Red Green, Red, Yellow Green, Yellow, Red What is the size of the sample space for choosing an ice cream flavor (chocolate, strawberry, vanilla) and sauce (chocolate, caramel, or none)? - Answ The answer is 9. From the list that was constructed, there are 9 outcomes. Vanilla, Chocolate Vanilla, Caramel Vanilla, None Chocolate, Chocolate Chocolate, Caramel Chocolate, None Strawberry, Chocolate Strawberry, Caramel Strawberry, None

What is the size of the sample space for spinning a color on a spinner labeled red, yellow, and blue and then spinning a number on a spinner labeled 1 through 4? - Answ 12 Any experiment or trial will have only one sample space. True or False? - Answ a. True Correct. This is a true statement. A sample space is the set of outcomes for one particular experiment, so any experiment has one unique sample space. All events in the sample space have the same relative frequency. True or False? - Answ b. False Correct. This is a false statement. The relative frequency of an event is how often it happens in practice. We do not know if people who do not have inherited risk factors for heart disease decrease their risk of developing heart-related diseases by eating healthy until we gather data. It is reasonable to assume, however, that the relative frequency of different eating habits (healthy versus not-healthy) might not be equal.

5. The sample space and sample size represent the same thing. True or False? - Answ b. False Correct. This is a false statement. The sample space is a set of outcomes while the sample size is the number of different outcomes.
6. The probability using sample spaces with equally likely outcomes is the number of outcomes in the event divided by the sample size. True or false?

• Answ a. True Correct. This is a true statement. The probability is the number of outcomes in the event divided by the sample size.

5. Finding the probability of events, given all outcomes are equally likely, has limited applications. True or False? - Answ a. True Correct. This is a true statement. Most probabilities are calculated when there are not equally likely outcomes.

Let A = Bob wears a black suit. Let B = Bob wears black shoes. We can now use this table to define some important categories in probability. Consider a man named Bob who has five suits and three pairs of shoes. He has one black suit, one brown suit, and three blue suits. He has two pairs of black shoes and one pair of brown shoes. We can put the two categories of clothes together. The table below lays out all the different possibilities of shoes and suits. - Answ universe "not" a and b intersectiion a or b union universe - Answ In our example of Bob's shoes and suits, the universe would be Bob's outfits, the ensemble of suits and shoes. These outfits would not include shirts, ties, or socks! Every cell in the table would be included in the universe. "not" - Answ Let A = Bob wears a black suit. Let B = Bob wears black shoes. We can now use this table to define some important categories in probability. In the table below, "not A" is illustrated in orange. Remember A = "Bob wears a black suit." "Not A " is Bob wears a blue or brown suit. The color of the shoes is irrelevant. In our Bob example, "not B " is not wearing black shoes. This is again illustrated in orange in the table below. a and b intersection - Answ For Bob's outfits, A and B would be the situations in which Bob is wearing both black suits and black shoes. A and B is represented below with orange cells. a or B union - Answ A or B is the category which includes all the outcomes in A, all the outcomes in B and all the outcomes in their intersection. It is called the union* of A or B. For Bob's outfits, A or B would be the situations in which Bob is wearing a black suit, black shoes, or both. A or B is represented below with orange cells.

venn diagrams - Answ For example, in the real world, we might ask what is the probability that a factory worker suffers a work-related injury in the past year? This question combines being a factory worker—a machinist, a textile worker, a structural iron and steel worker, or a mechanic (among many others)—and being a person who has suffered a work-related injury —a mailman slipping as he delivers mail, an assembler suffering repetitive stress injury, a construction worker falling from scaffolding (and many other injuries). The Venn diagram* can help convey the different categories. It shows a visual representation of all the possible results. The area labeled A represents A occurring—such as someone being a factory worker in the past year—while the area labeled B represents B happening—such being a person who suffered from a workplace injury in the past year. The darker area represents both A and B occurring, that is the set of all factory workers who suffered work-related injuries last year. THE MIDDLE IS WHERE U CAN SEE HOW MANY FACTORY WORKERS SUFFERED WORK RELATED INJURIES IN THE PAST YEAR universe venn diagram - Answ In our example of factory workers in the past year ( A ) and people who have suffered from workplace injuries in the past year ( B ), the universe would be "people who were employed in the past year." The universe is defined based on the question that researchers are interested in studying. If we were calculating probabilities such as the probability of factory workers being injured at work compared to the probability of other workers being injured at work, it would make sense for the universe to be "all employed people within the past year." In the diagram below, the orange shaded area represents all the elements in the universe. "not" venn diagram - Answ "Not A " in our real-world example is "Americans employed last year as something other than factory workers." Teacher, postal worker, librarian, business woman, etc. all belong to "not A ." Note that it includes people in category B who are not in A , such as librarians who were injured by falling bookshelves. However, it is best not to explicitly consider B when trying to decide if something is not A. It's enough to ask "is this person someone who was employed last year in a

type of factory work?". If not, she belongs to "not A ." In the diagram below, the orange shaded area represents all the elements in "not A ." Not B refers to outcomes that are not in the category of B but still in the universe. In our real-world example, not B would be employed people who were NOT injured at work last year, including all factory workers who were not injured at work, as well as all doctors, nurses, garbage workers, and truck drivers who were not injured at work. In the diagram below, the orange shaded area represents all the elements in "not B " a and b intersection venn diagram - Answ the middle of the venn diagram a or b union venn diagram - Answ the whole venn diagram the probabilities of each individual outcome in the sample space sum to 1. - Answ Similarly, if the experiment is flipping a coin, the events are "heads" or "tails." The probability of flipping a heads is 12. The probability of flipping a tails is 12. The sum of these two probabilities is 1/2 + 1/2= 1 complement - Answ the opposite of an event happening THE OCCURANCE OF THE EVENT NOT HAPPENING the sum = 1 Complementary events are those that do not have any common outcomes, and the union is the whole universe. One outcome is the event itself, and the other outcome is the event not happening. For example, flipping a coin and it landing on either heads or tails — those are the only two possible outcomes, so the two events are complementary. If there are outcomes in the universe that are not in either event, then the two events are not complementary. For example, picking either a large shirt or a medium shirt are not complementary events. There are shirts in many sizes! You could pick a large, medium, small, extra large, etc. So picking a large shirt or a medium shirt are not complementary.

The events must be properly defined in order to have complementary events. For example, "rolling a die and getting an even number" would not be a complete instance of complementary events (as this is only one event). However, "rolling a die and getting an even number" and "rolling the die and getting an odd number" are complementary events; the events are clearly defined. Notice from the example above that since the sum of the probability of an event and its complement is 1 , the probability of 1 minus the probability of that event is the probability of the complement. - Answ P(A)+P(not A)=11−P(A)=P(not A)1−P(not A)=P(A) 1 − P(complement) P(at least one X) = 1 − P(no X)

2. If two events have no common outcomes, they are complementary. True of False? - Answ b. False Correct. This is a false statement. Two events with no common outcomes are complementary only if their union is the whole universe. disjoint - Answ If two events cannot both occur at the same time, they are called disjoint. This concept applies to any number of multiple events. For example, a coin cannot be both "heads" and "tails" at the same time. The three statements below are disjoint: Examples of Disjoint Statements I was born in January. I was born in March. I was born in May. population and sample - Answ A population and a sample* can be disjoint as well, but they shouldn't be. For example, suppose you are studying the prevalence of obesity in pre-school children. You would not survey the local high school in order to gather data about the population of pre-schoolers. That would make your population and your sample disjoint, which is nonsensical if you're trying to use your sample to learn something about your population. On the other hand, taking two disjoint samples can make

sense: in your study of obesity in pre-schoolers, you might take disjoint samples—one in California, one in Massachusetts. disjoint a and b intersection - Answ It is relatively easy to calculate the probability of two or more disjoint events. If there is no overlap, there is no possibility of belonging to both sets. The probability that someone is born in both March and January, for example, is 0. This is true of the intersection of all disjoint events: If A and B are disjoint, P(A and B)= disjoint a or b union - Answ addition rule of probability If A and B are disjoint, P(A or B)=P(A)+P(B) What is the probability that someone is born in January or March, for example? The sample set of "month of birth" is comprised of the 12 months of the year. If you think about it, there are 2 ways to be born in January or March. You can be born in January ( 1 chance out of 12 months) or you can be born in March ( 1 chance out of 12 months). These 2 chances lead to a probability of 2/. disjoint and complementary - Answ Note that events can be disjoint* without being complementary*. An example of events that are both disjoint and complementary are "flipping heads" and "flipping tails." Together these events constitute the universe of possible events, so they are complementary. You also cannot have them at the same time, which means they're disjoint. But the events "being born on a Wednesday" and "being born on a Friday" are disjoint without being complementary. There are other possibilities so even if you are not born on a Wednesday, that does not necessarily mean you are born on a Friday. independent events - Answ are those that are not affected by other trials or events. For example, if you were to flip a coin once, that first result (either heads or tails) would not have any impact when the coin is flipped a second time — the first event gives no indication of what could result from the second event. A more real-world example is two people who don't know

each other having heart attacks. If someone has a heart attack, that has no effect on the probability the second person will have a heart attack. Example: An Event That is Not Independent - Answ In a candy container, there are five white chocolates and ten dark chocolates. If you pick a chocolate at random out of the jar, the probability of what you will get on the next pick is affected. If you pick a dark chocolate on your first pick, there will be fewer dark chocolates the next time you pick, lowering the probability you get a dark one on the second draw. On the other hand, if you pick a white chocolate on your first pick, there will be fewer white chocolates the next time you pick, lowering the probability you will get a white on the second draw. Unlike the coin toss, what you get in the first trial is not independent from what you get in the second trial. Another, more real world example, is the event "smoking cigarettes" and "getting lung cancer." The occurrence of the first event affects the probability of the second event. Disjoint Events: Dependent - Answ disjoint events are dependent Disjoint events are dependent. For example, if you know someone is born on a Wednesday, you also know that he or she cannot be born on a Friday. If you know someone is not born on a Wednesday, there is a greater possibility, 16 rather than 17 , that he or she is born on a Friday. Because the occurrence of one affects the probability of the other, these events are dependent. multiplication rule for independent events - Answ P(A and B)=P(A)×P(B) what is the chance of rolling double 1s when rolling 2 dice? 1/6 x 1/6= 1/ to calculate an "or" problem - Answ add the probability of both events. The probability of one of two (or more) events occurring is just the probability of each event added together. But notice that when you add all the times Bob wears black shoes and all the times Bob wears a black suit, you are counting the cells where both occur twice. P(A or B)=P(A)+P(B)−P(A and B)

subtract the intersection dependent examples - Answ The probability of picking a card that is a heart, given holding three hearts in your hand. The probability of flipping a coin and getting three heads in a row, given an initial flip of heads. The probability of getting in an accident, given a certain length of daily commute. The probability that your car is red, given that your previous car was red. The probability of getting lung cancer given a history of smoking. The probability of being a gifted athlete given that one biological parent was a gifted athlete. independent examples - Answ The probability of flipping a tails, given the previous flip was heads The probability that you are born in March, given that your spouse is born in March. The probability of drawing an ace after drawing an ace and then putting it back in the deck. conditional probability - Answ dependent events is the probability of an event occurring (such as picking the winning ticket), given that another event has already occurred (such as someone else having already picked one of the losing tickets). P (B given A)= p (A and B)/ p (A) If A and B are independent events, P(B|A)=P(B). The probability of going fishing on a Saturday is represented as P(F). The probability of catching a fish on a Saturday is P(C). How would we write the probability of catching a fish on a Saturday, given that you have gone fishing? - Answ P(C|F) The General Multiplication Rule - Answ P(B∣∣∣A)=P(A and B)/ P(A)

we can multiply both sides by pA to solve for p A and B P(A and B)=P(A)×P(B|A) 26% of Americans age 65 or older have diabetes. 68% of people over 65 with diabetes die from heart disease. What is the probability of having diabetes and dying from heart disease? - Answ Let A= being 65 or older with diabetes Let B= dying from heart disease after age 65 P(A)=26% P(B|A)=68% What is the P(A and B)? P(A and B)=P(A)×P(B|A)P(A and B)=0.26×0.68=0. 2 events are independent if - Answ P(A and B)=P(A)⋅P(B) P(A|B)=P(A) P(B|A)=P(B) P(B|A)=P(B| not A) Eight toys are being given out to eight children in a hospital ward: a green boat a blue boat a pink sparkly fairy doll a blue sparkly fairy doll a yellow toy telephone a blue whale plush animal a blue light-up star a pink light-up star - Answ t A= "getting a pink item" and B= "getting a boat." Are A and B independent events? Let us select one of the equalities above to determine if A and B are independent events. Calculate P(A|B) and P(A). P(A|B)= B is given, so Ari has gotten a boat. The boats don't come in pink, so the probability that Ari gets a pink item is 0. P(A)=1/

There are 2 pink items and 8 items total. P(A|B)≠P(A) , therefore the events are not independent. 1/4 does not equal 0 Let A= "getting a blue item" and B= "getting a boat" Are A and B independent events? Let us use P(A and B)=P(A)⋅P(B) to test for independence. P(A and B)= There is one item that is both blue and a boat. The intersection of "getting a blue item" and "getting a boat" is the blue boat. P(A)= There are 4 blue items and 8 items total. P(B)= P(A)⋅P(B)=12×14=1/ Therefore P(A and B)=P(A)⋅P(B) and the events are independent A= "getting a pink item" and let B= "getting a fairy doll" Are A and B independent events?

3. For independent events, what does P(A and B) equal? - Answ P(A)⋅P(B) A and B are independent events. P(A)=1 and P(B)=0.1. Calculate P(A and B). - Answ 0. A and B are independent events. P(A)=0.3 and P(B)=0.8. Calculate P(A|B).

• Answ 0. A and B are independent events. P(A)=0.5 and P(B)=1. Calculate P(B|A).
• Answ 1 law of total probability - Answ The fact that if you multiply across each branch and then add up the resulting products, you get 1

1. A probability tree has the same information as a tree diagram. True or False? - Answ False

Correct. This is a false statement. A probability tree contains the probability of an event, not just which events occur. The law of total probability is the sum of the product of all sequences of possible outcomes in a sample space is 1. True or False? - Answ a. True Correct. This is a true statement. In a probability tree, multiply each branch of the tree and the sum of the products is 1 A game has a 10-sided die. What is the probability of rolling an even number or a number less than 5? - Answ The probability of an even number is fraction begin. numerator: 5 denominator: 10 fraction end. and rolling a number less than 5 is fraction begin. numerator: 4 denominator: 10 fraction end.. There are 2 values that are even and less than 5 (2 and 4). The probability is fraction begin. numerator: 5 denominator: 10 fraction end. + fraction begin. numerator: 4 denominator: 10 fraction end. - fraction begin. numerator: 2 denominator: 10 fraction end. = fraction begin. numerator: 9 denominator: 10 fraction end. - fraction begin. numerator: 2 denominator: 10 fraction end. = fraction begin. numerator: 7 denominator: 10 fraction end.. There are black, blue, and white marbles in a bag. The probability of choosing a black marble is 0.36. The probability of choosing a black and then a white marble is 0.27. To the nearest hundredth, what is the probability of the second marble being white if the first marble chosen is black? - Answ Use P(B | A) = fraction begin. numerator: P(A and B) denominator: P(A) fraction end.. The probability of black and white is divided by the probability of black. Therefore,

fraction begin. numerator: 0. denominator: 0. fraction end. = 0.75. A man will have enough money to pay the rent if he gets 100 hours of work at job A or if his wife earns enough money to pay the rent for both of them. If we want to find the probability that this couple will be able to pay the rent, What is the relevant piece of information to know when deciding whether to use the formula P(A or B) =P(A)+P(B) or P(A or B) =P(A)+P(B)−P(A and B) ? - Answ a) Whether there is any possibility of his wife earning enough money and his getting 100 hours of work at job A .We must know if the two events are disjoint or not. Roger has two blue ties, one red tie, two blue coats, and two orange coats. What is the probability that he wears either a blue tie or a blue coat? - Answ 5/ 2/3 + 1/2 - (A and B = 1/3) - 1/3= 5/ At a restaurant for lunch, there are 4 sandwich choices, 6 side choices, and 8 drink choices. What is the size of the sample space if one choice is selected from each? - Answ d) 192 Correct. The Answ is d. There are 4×6×8=192 choices in the sample space. For each day in New England, the probability that it is mostly cloudy is 0.

. Cindi is planning to go leaf watching, and her friend will join her with probability 0.70. Find the probability that when Cindi takes her trip, either Cindi's friend does not come to New England, or she does come but it is mostly cloudy. Assume that the timing of her coming to New England is independent of the weather conditions. - Answ b) 0. Correct. The Answ is b. The probability that the friend doesn't come is 0.. The probability that she comes and it's cloudy is 0.7×0.1=0.07. The probability of one or the other of these events is 0.3+0.07=0..