Probability and Statistics Exercises: 120 Questions and Solutions, Exams of Statistics

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C784 MODULE 7 (120 QUESTIONS AND SOLUTIONS 100 % CORRECT) 2024-2025 UPDATED GRADED 100% PASS (ACTUAL EXAM) PROBABILITY - Solution it is the chance of an event occurring PERCENTAGE - Solution parts of a whole Impossible - Solution 0% possibility Probability that the day of the month will be 32 Unlikely - Solution 1% to 30% probability A snowstorm in Boston leaves more than 15 inches of snow As likely as unlikely - Solution 40% to 60% probability A pregnant woman having a boy rather than a girl Likely - Solution 70% to 99% probability The probability that a child born in the United States will live to adulthood Certain - Solution 100% probability The probability that everyone alive today has a birthday Decimal Expression of Percentage - Solution probability percentages range strictly from 0% to 100% Therefore the decimal form of percentages range from 0 to 1 (remember to obtain the decimal form of a percentage, you divide by 100) EXPERIMENT - Solution is the procedure, or situation, for which the probability is being calculated. So the roll of a die can be an experiment, a coin flip can be an experiment, or even the weather on a particular Wednesday OUTCOME - Solution The possibilities of what can occur during an experiment—the results of the experiment— When rolling a die, 1, 2, 3, 4, 5, and 6 are the possible outcomes EVENT - Solution is comprised of one or more outcomes. The die landing on an even number is an event. The die landing on 1 is also an event FAIR - Solution if each outcome is equally likely. P(E) - Solution The probability of an event is represented by P(E) which means probability (P) of a certain event (E) occurring. If R = Rain and the weatherman says there is a 40 percent chance of rain, then P(R) = 0.40. (Notice that the probability is written in its decimal form). . An experiment is the procedure to test the occurrence of an event. True or False? - Solution FALSE An experiment is the procedure for which the probability of an event is calculated. The possible results of an experiment are outcomes. True or False? - Solution TRUE Each of the possible results from an experiment is known as an outcome. The probability of an event occurring can be greater than 100%. True or False? - Solution false The probability of an event is always between 0% and 100%. 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? - Solution TRUE 10,000 tosses is a large number. By the law of large numbers we would expect the relative frequency to be close to 1/6, the theoretical probability. After a certain number of trials the empirical probability of an outcome will equal the theoretical probability. True or False? - Solution FALSE 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? - Solution 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. 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.) Kiki the dog has a yellow, blue, green, and orange shirt. If her owner reaches into the drawer and picks one out at random, what is the probability the shirt is NOT orange? - Solution 3/4 When rolling a fair, six-sided die, what is the probability of rolling a 2? - Solution 1/6 There is one outcome that corresponds with the desired event, out of six total possible outcomes If you have a full, standard deck of cards, what is the probability of selecting a queen of hearts? - Solution 1/52 You have a jar of 7 marbles: 1 red marble, 2 green marbles, and 4 blue marbles. Selecting a marble at random, what is the probability that you select a green marble? - Solution 2/7 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? Theoretical probability? Empirical Probability? The law of large Numbers? - Solution Empirical Probability We are using the number of times an event occurred to estimate the probability, therefore this is an example of empirical probability. The Cardinals have won 100 of their 162 games. Using only this data, what is the probability that the Cardinals win a game? - Solution 100/162 50/81 62% When rolling a fair, six-sided die, what is the probability of rolling a 2 or a 3? - Solution 1/3 There are two outcomes that correspond with the desired event (rolling a two or a three), out of six total number of outcomes that can occur. 2/6 This fraction simplifies TO 1/3 You examine a deck of cards to determine the probability of selecting a certain group of cards. What is this an example of? Theoretical probability? Empirical Probability? The law of large Numbers? - Solution Theoretical probability We are using the number of times an outcome would occur, and they are all equally likely, this is an example of theoretical probability. SAMPLE SPACE - Solution The set of possible outcomes in an experiment The sample space of rolling a regular six-sided die is 1, 2, 3, 4, 5, and 6 SAMPLE SIZE - Solution is the number of different outcomes the sample size of rolling a six-sided die is 6. It is important to remember that the probabilities of all the outcomes in the sample space always sum to 1. - Solution That is, there is a 100 percent chance of one of the outcomes in the sample space happening, since the sample space, by definition, contains all possible outcomes. What is the sample space for flipping 3 coins? - Solution The answer is 8. 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 sample space for rolling 1 dice and flipping 1 coin? - Solution The answer is 12. From the list that was constructed, there are 12 outcomes. 1, Heads 1, Tails 2, Heads 2, Tails 3, Heads 3, Tails The probability of an event can never be greater than 1. 4. A tree diagram is the only method to determine the probability of an event. True or False? - Solution FALSE A list, table, tree diagram, and other methods can help determine the probability of an event. 5. Finding the probability of events, given all outcomes are equally likely, has limited applications. True or False? - Solution TRUE Most probabilities are calculated when there are not equally likely outcomes. Won Lost Won W, W W, L Lost L, W L, L What is the probability that you won both rounds of rocks-paper-scissors- shoot? - Solution 1/4 There are four equally likely outcomes. One of those outcomes involves winning both rounds Won Lost Won W, W W, L Lost L, W L, L What is the probability that you lost both rounds of rocks-paper-scissors- shoot? - Solution 1/4 There are four equally likely outcomes. One of those outcomes involves losing both rounds Won Lost Won W, W W, L Lost L, W L, L What is the probability that you each win one round of rocks-paper- scissors-shoot, necessitating a third round? - Solution 1/2 here are four equally likely outcomes. Two of those outcomes involves you each winning one round of rocks-paper-scissors-shoot Won Lost Won W, W W, L Lost L, W L, L What is the probability that you do NOT win both rounds of rocks-paper- scissors-shoot? - Solution 3/4 here are four equally likely outcomes. Three of those outcomes do NOT involve you winning both rounds Won Lost Won W, W W, L Lost L, W L, L While this table shows the sample space after two rounds, how large would the sample space be after three rounds, assuming all three rounds are played regardless of the outcome of the first two rounds? - Solution 8 The sample space after three rounds would be 8. WWW LLL WWL LLW WLW LWL WLL LWW Won Lost Won W, W W, L Lost L, W L, L While this table shows the sample space after two rounds, how large would the sample space be after just one round? - Solution 2 The sample space after one round would be 2. You roll a six-sided die. What is the size of the sample space after one die roll? - Solution 6 The size of the sample space of the possible outcomes after one roll is 6. There are six possible outcomes. You roll a six-sided die twice. What is the size of the sample space after those two rolls? - Solution 36 The size of the sample space of the possible outcomes after two rolls is 36. There are 36 possible outcomes. You roll a six-sided die twice. What is the probability of rolling two sixes? - Solution 1/36 The size of the sample space of the possible outcomes after two rolls is 36. These outcomes are equally likely. There is one outcome that results in two sixes 4) You roll a six-sided die twice. What is the probability of rolling the same number both times? - Solution 1/6 The size of the sample space of the possible outcomes after two rolls is 36. These outcomes are equally likely. There are six outcomes that results in rolling the same number both times [(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)]. 5) You roll a six-sided die three times. What is the size of the sample space after those three rolls? - Solution 216 The size of the sample space of the possible outcomes after three rolls is 216. There are 216 possible outcomes. 6) You roll a six-sided die three times. What is the probability of rolling all three 1s? - Solution 1/216 The size of the sample space of the possible outcomes after three rolls is 216. These outcomes are equally likely. There is one outcome that results in rolling all three 1s. 3. If the probability of the event given is 0, then the complementary event will have probability equal to 1. True or False? - Solution TRUE The sum of the probabilities of the event and its complement is 1, so if the probability of the given event is zero, its complement has probability equal to 1. DISJOINT - Solution If two events cannot both occur at the same time, they are called disjoint Examples of Disjoint Statements I was born in January. I was born in March. I was born in May. POPULATION - Solution An entire pool from which a sample is drawn. Samples are used in statistics because of how difficult it can be to study an entire population SAMPLE - Solution a group of people chosen to represent a larger population Disjoint A and B (intersection) - Solution 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) = 0 Disjoint A or B (union) - Solution Since there is no overlap, the probability of being in either one or the other is the sum of their individual probabilities. What is the probability that someone is born in January or March, for example? 2/12 OR 1/6 ADDITION RULE OF PROBABILITY - Solution for disjointed events A and B, P(A or B) = P(A) + P(B) A 3 year old weighs at least 25 lbs and the same 3 year old weighs 27 lbs. disjoint or non-disjoint - Solution NON DISJOINT a3 year old can weigh 27 lb and can weigh over 25 lbs A six-sided die is showing a 5 and a 1 disjoint or non-disjoint - Solution DISJOINT Only a 5 or a 1 can be showing on the die, not both A number shown on a die is even and a number is less than 3 disjoint or non-disjoint - Solution NON DISJOINT 2 is even and 2 is less than 3, so these events are not disjoint 1. When researching or trying to learn something about a given population, the population and the sample should always be disjoint. True or False? - Solution FALSE While a population and a sample can be disjoint, they shouldn't be if you're trying to use your sample to learn something about your population! 2. A sample space S is comprised of 13 equally likely outcomes. Suppose event E contains 5 of those outcomes. The probability of E is 5/13 True or False? - Solution TRUE Correct. This is a true statement. Each event has a 1⁄13 probability of occurring, so one of 5 occurring is 5/13 3. The addition rule for disjoint events would yield the same result as the general formula for theoretical probability, counting the ways of getting a certain result. True or False? - Solution TRUE We have done problems both ways in this module. We add the individual probabilities for the disjoint events of, say, being born in February OR being born in April. Or we count up all the "ways" of being born in either February or April and put that number over the total size of the sample space. Either method for this problem would yield 2/12 = 1/6 4. If 2 events can occur at the same time, they are called disjoint. True or False? - Solution FALSE Disjoint events are 2 events that cannot occur at the same time. 5. If two events are disjoint there is no intersection between the events. True or False? - Solution TRUE There are no common elements between 2 disjoint events. INDEPENDENT EVENT - Solution Independent events* 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) INDEPENDENT - Solution events, where the occurrence of one does not affect the probability that the other event will occur Example: An Event That is Not Independent - Solution 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 Disjoint Events: Dependent - Solution Disjoint events are dependent. P(A) = 0.3 P(B) = 0.2 P(A and B) = 0.3 x 0.2 6% of the time, she will be caught in traffic dependent Events - Solution the probability *of getting lung cancer given a history of smoking *that your car is red, given that your previous car was red *your good at sports given that one parent was good at sports *picking a card that is a heart, while holding 3 hearts in your hand Independent events - Solution the probability *you are born in march, given that your spouse is born in march *flipping tails given that you just flipped heads 1. A coin is flipped three times. What is the probability of flipping three heads in a row (do NOT construct a tree; instead, use the multiplication principle)? - Solution 1/8 There are 2 possibilities for each coin toss. The probabilities are 1/2 * 1/2 *1/2 = 1/8 2. A puppet-making booth has the choice of 3 different color bodies and 3 types of head. What is the probability of making one particular type of puppet? - Solution 1/9 There are 3 body types and 3 types of head, there are 9 different kinds of puppets. 1/3 *1/3 *1/3 = 1/9 3. In the problem above, making a puppet with a red body and making a puppet with a blue body are independent events. True or False? - Solution FALSE You can only choose one color for the body of the puppet, so choosing a red-bodied puppet and choosing a blue-bodied puppet are disjoint events. Sally has 3 coats - one is blue, one is red, and one is green. She also has red and green gloves. Everyday she wears one coat and one set of gloves. Which of the following uses the General Addition Rule to calculate the probability that Sally wears green on any given day? (The fractions are unsimplified to make calculation easier.) - Solution 2/6 + 3/6 - 1/6 The General Addition Rule gives us a way of calculating the probability of two independent events. (RC, RG) (GC, RG) (BC, RG) (RC, GG) (GC, GG) (BC, GG) Notice that the probability of Sally wearing a green coat is 2 out of 6 cells in the table. The probability that she wears green gloves is 3 out of 6 cells in the table, but we have double counted wearing both a green coat and green gloves, so we must subtract the one cell in which she is wearing both green items. 5. Events that do not influence each other are independent. True or False? - Solution Independent events are events that do not affect each other. 1. A frequency table consists of two categorical variables. True or False? - Solution TRUE A frequency (or two-way) table contains two different categorical variables. 2 A frequency table is read by the row and column entries. True or False? - Solution TRUE Each entry is read by the row and column entry. 3. The only information you need to perform analysis in a two-way table can be provided by row entries. True or False? - Solution FALSE You also need the totals in columns and rows to perform analysis. 4. Relative frequency helps determine if an event is independent. True or False? - Solution The relative frequency is related to the independence of events. 5. The frequency table does not need labels for rows or columns. True or False? - Solution FALSE A frequency table needs labels for rows and columns to display the data accurately.